Electricity — Notes

Electric Current and Circuit

Electric current is the flow of electric charge through a conductor. Just as a river current is flowing water, an electric current is flowing electrical charge. In metallic wires, the charges that flow are electrons.

An electric circuit is a continuous and closed path through which electric current can flow. For example, in a torch, the cell, bulb, and switch form an electric circuit. The switch must be "on" to complete the circuit and allow current to flow. If the circuit is broken anywhere (like turning the switch "off"), the current stops.

Measuring Electric Current

Electric current is defined as the rate of flow of electric charge. It is quantified by the amount of charge passing through a specific area of a conductor per unit of time.

Mathematically, if a net charge $Q$ flows across a cross-section of a conductor in time $t$, the current $I$ is given by:

• Unit of Charge: The SI unit of electric charge is the coulomb (C). One coulomb is approximately the charge of $6\times {10}^{18}$ electrons. An electron carries a charge of $-1.6\times {10}^{-19}$ C.

• Unit of Current: The SI unit of electric current is the ampere (A), named after scientist Andre-Marie Ampere. One ampere is defined as one coulomb of charge flowing per second:

Smaller units are milliampere (mA) ($1\text{ mA}={10}^{-3}\text{ A}$) and microampere ($\mu$A) ($1\mu \text{A}={10}^{-6}\text{ A}$).

• Measurement: An instrument called an ammeter is used to measure electric current in a circuit. It is always connected in series with the component through which the current is to be measured.

Direction of Electric Current

• Conventional Current: At the time electricity was first studied, the flow of positive charges was understood. Therefore, the direction of electric current was conventionally defined as the direction of flow of positive charges.

• Electron Flow: In most metallic conductors used in circuits, the charge carriers are electrons, which are negatively charged. Electrons flow from a region of lower potential to a region of higher potential.

• Relationship: The conventional direction of electric current is taken to be opposite to the direction of flow of electrons.

Example 11.1: A current of 0.5 A is drawn by a filament of an electric bulb for 10 minutes. Find the amount of electric charge that flows through the circuit.

Given: Current, $I=0.5$ A Time, $t=10$ minutes

To find: Amount of electric charge, $Q$.

Solution: First, convert time to seconds: $t=10\text{ min}\times 60\text{ s/min}=600$ s

Using the formula $I=\frac{Q}{t}$, we rearrange to find $Q$:

$Q=300$ C

Therefore, the amount of electric charge that flows through the circuit is 300 coulombs.

Remember

The direction of conventional current is from positive to negative, opposite to the flow of electrons. 🎯 Exam-smart: You might be given the current and time and asked to find the charge, or given charge and time to find current. Always ensure time is in seconds.

Calculating Number of Electrons in One Coulomb

To find the number of electrons ($n$) that constitute one coulomb of charge ($Q$), we use the relationship $Q=n\times e$, where $e$ is the charge of one electron.

Given: $Q=1$ C Charge of one electron, $e=-1.6\times {10}^{-19}$ C (we use the magnitude for counting)

Rearranging the formula:

So, approximately $6.25\times {10}^{18}$ electrons constitute one coulomb of charge.

Electric Potential and Potential Difference

The flow of electric charge in a conductor is not spontaneous; it requires a driving force, analogous to how water flows only when there's a difference in pressure or height. In electric circuits, this driving force is the potential difference.

What is Potential Difference?

Potential difference is the work done per unit charge to move a charge between two points in an electric field. It is often referred to as voltage.

• Analogy: Imagine water flowing through a pipe. Water flows from a higher level to a lower level due to a pressure difference. Similarly, electric charges flow through a conductor if there is a difference in electric pressure, which is the potential difference.
• Source: This potential difference is typically maintained by a battery or an electric cell. Chemical reactions within a cell create a potential difference across its terminals.
• Maintaining Current: When a conductor is connected to a cell, the potential difference causes charges to move, establishing an electric current. The cell continuously expends its stored chemical energy to maintain this potential difference and keep the current flowing.

Definition and Formula

The potential difference (V) between two points in an electric circuit carrying current is defined as the work done (W) to move a unit charge (Q) from one point to the other.

Where:

• $V$ is the Potential Difference (in Volts, V)
• $W$ is the Work Done (in Joules, J)
• $Q$ is the Charge (in Coulombs, C)

Unit of Potential Difference

The SI unit of electric potential difference is the volt (V), named after Alessandro Volta.

• Definition of 1 Volt: One volt is the potential difference between two points in a current-carrying conductor when 1 joule of work is done to move a charge of 1 coulomb from one point to the other.

Measurement of Potential Difference

The instrument used to measure potential difference is called a voltmeter.

• Connection: A voltmeter is always connected in parallel across the two points between which the potential difference is to be measured. This ensures it measures the potential drop across that component without significantly altering the circuit's current.

Example Application

Problem: How much work is done in moving a charge of 2 C across two points having a potential difference of 12 V?

Solution: We are given:

• Charge, $Q=2$ C
• Potential Difference, $V=12$ V

Using the formula $V=\frac{W}{Q}$, we can find the work done, $W$:

Therefore, 24 joules of work are done.

Remember

Potential difference is the driving force for electric current. Without it, charges would not move.

Exam-smart

Questions often provide work done and charge, asking for potential difference, or provide potential difference and charge, asking for work done. They might also provide potential difference and work done, asking for charge. Always ensure units are consistent (Volts, Joules, Coulombs).

Circuit Diagram

An electric circuit is a complete, closed path through which electric current can flow. It typically consists of a power source (like a cell or battery), connecting wires, and one or more electrical components (like a resistor or a bulb). To represent these circuits easily, we use circuit diagrams which employ standardized symbols for each component. This makes it simpler to understand, draw, and analyze complex electrical arrangements without needing to show the actual physical components.

Symbols Used in Circuit Diagrams

Different electrical components are represented by specific symbols in a circuit diagram. This standardization allows anyone familiar with these symbols to understand the circuit's layout and function.

Remember

The positive terminal of a cell or battery is represented by the longer line in its symbol, and the negative terminal by the shorter line.

Exam-smart

In a circuit diagram question, you might be given a diagram with specific symbols and asked to identify the components or describe the circuit's function. Alternatively, you might be asked to draw a circuit diagram based on a description, ensuring you use the correct symbols for components like cells, resistors, and switches. For instance, a question could describe a circuit with a battery, a switch, and a bulb, and you'd need to draw it using the appropriate symbols, showing how they are connected.

Ohm’s Law

Ohm's law establishes a fundamental relationship between the potential difference ($V$) across a conductor and the electric current ($I$) flowing through it. It states that for a given metallic conductor at a constant temperature, the potential difference across its ends is directly proportional to the current flowing through it.

Mathematically, this can be expressed as:

This proportionality means that if you increase the potential difference, the current will increase by the same factor, and vice versa, provided the temperature of the conductor remains unchanged.

The constant of proportionality in this relationship is called resistance, denoted by $R$. Therefore, Ohm's law can be rewritten as:

This equation is a cornerstone of electrical circuit analysis. It allows us to calculate any one of the three quantities ($V$, $I$, or $R$) if the other two are known.

The resistance ($R$) of a conductor is a measure of its opposition to the flow of electric current. It quantifies how much the conductor restricts the movement of electrons. The SI unit of resistance is the ohm, symbolized by the Greek letter Omega ($\mathrm{\Omega }$).

From the equation $V=IR$, we can also define resistance as:

This means that if a potential difference of 1 volt ($1$ V) across the ends of a conductor causes a current of 1 ampere ($1$ A) to flow through it, then the resistance of the conductor is 1 ohm ($1$ $\mathrm{\Omega }$).

The relationship $I=\frac{V}{R}$ also highlights that the current flowing through a conductor is inversely proportional to its resistance. If the resistance of the conductor doubles, the current is halved, assuming the potential difference remains constant. This is because a higher resistance impedes the flow of charge more effectively.

Variable Resistance (Rheostat)

In many electrical circuits, it is necessary to adjust or regulate the current. A component used for this purpose, which allows for changing the resistance in the circuit without altering the voltage source, is called variable resistance. A common device that functions as a variable resistance is a rheostat. Rheostats are used to control the amount of current flowing through a circuit, for instance, to dim a light bulb or control the speed of a motor.

Remember

Ohm's law is only valid when the temperature of the conductor remains constant. Any significant change in temperature can affect the resistance and thus the linear relationship between $V$ and $I$.

Exam-smart

In competency-based questions, you might be given a set of readings for potential difference ($V$) and current ($I$) for a conductor. You will be asked to verify Ohm's law, calculate the resistance, or predict the current/potential difference for a new set of values based on the established proportionality. You might also see a V-I graph, where a straight line passing through the origin indicates that Ohm's law is obeyed.

Factors on Which the Resistance of a Conductor Depends

The resistance ($R$) of a conductor is a measure of its opposition to the flow of electric current. It quantifies how much the conductor "hinders" the movement of charge carriers (usually electrons). A higher resistance means a greater opposition to current flow.

Several factors influence the resistance of a conductor:

1. Length of the Conductor ($l$)

Resistance is directly proportional to the length of the conductor. This means if you double the length of a wire, its resistance also doubles.

• Why? A longer wire provides a longer path for the electrons to travel. As electrons move through the conductor, they collide with the atoms of the material. In a longer wire, there are more atoms and thus more opportunities for collisions, leading to greater opposition to the flow of current.
• Analogy: Imagine trying to push a ball through a long, narrow pipe versus a short, narrow pipe. The ball will face more resistance and slow down more in the longer pipe.

Mathematically, this relationship is expressed as:

2. Area of Cross-Section ($A$)

Resistance is inversely proportional to the area of cross-section of the conductor. This means if you double the cross-sectional area (make the wire thicker), the resistance halves.

• Why? A thicker wire provides more pathways for the electrons to flow simultaneously. With a larger area, there are more charge carriers available to move, and they experience less obstruction.
• Analogy: Imagine water flowing through a pipe. Water flows more easily through a wide pipe than through a narrow pipe.

Mathematically, this relationship is expressed as:

3. Nature of the Material

Different materials have different inherent abilities to resist the flow of electric current. This property is called resistivity.

• Resistivity ($\rho$): This is a characteristic property of a material that measures how strongly it resists electrical current. It is independent of the shape and size of the conductor. Materials with low resistivity are good conductors, while those with high resistivity are poor conductors or insulators.
• Metals and Alloys: Have very low resistivity (typically in the range of ${10}^{-8}\text{ to }{10}^{-6}\mathrm{\Omega }\text{ m}$). They are good conductors.
• Insulators: Have very high resistivity (typically in the range of ${10}^{10}\text{ to }{10}^{17}\mathrm{\Omega }\text{ m}$). They are poor conductors.

4. Temperature

The resistance of most conductors generally increases with an increase in temperature.

• Why? At higher temperatures, the atoms in the conductor vibrate more vigorously. This increased atomic motion leads to more frequent collisions with the moving electrons, thus increasing the resistance.
• Note: For some materials like semiconductors, resistance decreases with increasing temperature. However, for metallic conductors commonly used in circuits, resistance increases with temperature.

Combining the Factors: The Formula for Resistance

By combining the relationships for length and area of cross-section, we can write the resistance of a uniform metallic conductor as:

Where:

• $R$ is the resistance of the conductor (in Ohms, $\mathrm{\Omega }$).
• $\rho$ (rho) is the electrical resistivity of the material (in Ohm-meters, $\mathrm{\Omega }\text{ m}$).
• $l$ is the length of the conductor (in meters, m).
• $A$ is the area of cross-section of the conductor (in square meters, ${\text{m}}^2$).

Resistivity is a fundamental property of the material. For example, copper has a lower resistivity than iron, making it a better conductor.

Properties of Alloys vs. Pure Metals

Real-World Applications

• Filaments of Electric Bulbs: Made of Tungsten, which has a high melting point and suitable resistivity to glow brightly when heated by the current.
• Heating Elements: Made of Nichrome (an alloy of Nickel, Chromium, Manganese, and Iron) because of its high resistivity and ability to withstand high temperatures without melting or oxidizing.
• Electrical Transmission Lines: Made of Copper or Aluminium because they have very low resistivity, minimizing energy loss as heat during transmission.

Remember

Resistivity ($\rho$) is a property of the material itself, while resistance ($R$) depends on the material, length, and cross-sectional area of the specific object.

Exam-smart

You might be given the dimensions (length, diameter) and material of a wire, and asked to calculate its resistance. You’ll need to use $A=\pi r^2=\frac{\pi d^2}{4}$ and look up the resistivity from a table. Conversely, you might be given resistance and dimensions, and asked to find resistivity, then identify the material.

Resistance of a System of Resistors

We often combine resistors in electrical circuits to achieve a desired current or voltage. Understanding how these combinations behave is crucial for designing and analyzing electrical gadgets. There are two primary ways to connect resistors: in series and in parallel.

Resistors in Series

When resistors are connected end to end, such that the same current flows through each resistor, they are said to be connected in series. Imagine a single path for the electric current to flow – if you place multiple resistors along this single path, they are in series.

Key Characteristics of Series Combination:

• Current: The current is the same through all resistors in a series combination. If the total current flowing into the series combination is $I$, then the current through each resistor ($R_1$, $R_2$, $R_3$, ...) will also be $I$.
• Potential Difference: The total potential difference across the series combination is equal to the sum of the potential differences across each individual resistor. If $V_1$, $V_2$, and $V_3$ are the potential differences across resistors $R_1$, $R_2$, and $R_3$ respectively, and $V$ is the total potential difference across the combination, then $V=V_1+V_2+V_3$.
• Equivalent Resistance: The equivalent resistance (or resultant resistance) of a series combination is equal to the sum of the individual resistances. This means you can replace all the resistors in series with a single resistor of this equivalent value, and the circuit behavior will be the same.

Let's derive the formula for equivalent resistance in series. Consider three resistors $R_1$, $R_2$, and $R_3$ connected in series to a battery providing a potential difference $V$. The same current $I$ flows through each resistor.

Using Ohm's Law ($V=IR$) for each resistor: Potential difference across $R_1$ is $V_1=I{R}_1$. Potential difference across $R_2$ is $V_2=I{R}_2$. Potential difference across $R_3$ is $V_3=I{R}_3$.

The total potential difference $V$ across the combination is the sum of these individual potential differences:

Substituting the expressions from Ohm's Law:

We can factor out the common current $I$:

Now, let $R_s$ be the equivalent resistance of this series combination. According to Ohm's Law, the total potential difference $V$ is related to the total current $I$ and the equivalent resistance $R_s$ by:

Equating the two expressions for $V$:

Dividing both sides by $I$ (since $I\mathrm{\ne }0$):

For $n$ resistors connected in series, the equivalent resistance is:

Implication: The equivalent resistance in a series combination is always greater than the largest individual resistance. This is because you are adding resistances together, effectively increasing the total opposition to current flow.

Example Application: When you need to increase the total resistance in a circuit, you connect resistors in series. This is useful for controlling the current flow to a specific level. For example, old-style decorative Christmas lights often had bulbs connected in series. If one bulb burned out, the entire string would go dark because the circuit was broken.

Remember

In series, current is constant, and voltage divides. The equivalent resistance is the sum of individual resistances and is always larger than any single resistance.

Exam-smart

If given multiple resistors and asked to achieve the highest possible total resistance, connect them in series. In problems, if a circuit diagram shows components connected one after another in a single loop, they are in series.

Resistors in Parallel

When resistors are connected across the same two points, such that the potential difference across each resistor is the same, they are said to be connected in parallel. Imagine multiple paths for the electric current to split and flow through different components before rejoining.

Key Characteristics of Parallel Combination:

• Potential Difference: The potential difference is the same across all resistors connected in parallel. If the potential difference across points X and Y is $V$, then the potential difference across each resistor ($R_1$, $R_2$, $R_3$, ...) will also be $V$.
• Current: The total current flowing into the parallel combination is equal to the sum of the currents flowing through each individual branch (resistor). If $I_1$, $I_2$, and $I_3$ are the currents through resistors $R_1$, $R_2$, and $R_3$ respectively, and $I$ is the total current, then $I=I_1+I_2+I_3$.
• Equivalent Resistance: The reciprocal of the equivalent resistance of a parallel combination is equal to the sum of the reciprocals of the individual resistances.

Let's derive the formula for equivalent resistance in parallel. Consider three resistors $R_1$, $R_2$, and $R_3$ connected in parallel between points X and Y. A battery supplies a total current $I$ which splits into $I_1$, $I_2$, and $I_3$ through the respective resistors. The potential difference $V$ across each resistor is the same.

Using Ohm's Law ($I=V\mathrm{/}R$) for each resistor: Current through $R_1$ is $I_1=V\mathrm{/}{R}_1$. Current through $R_2$ is $I_2=V\mathrm{/}{R}_2$. Current through $R_3$ is $I_3=V\mathrm{/}{R}_3$.

The total current $I$ entering the junction X is the sum of the currents flowing through each branch:

Substituting the expressions from Ohm's Law:

We can factor out the common potential difference $V$:

Now, let $R_p$ be the equivalent resistance of this parallel combination. According to Ohm's Law, the total current $I$ is related to the total potential difference $V$ and the equivalent resistance $R_p$ by:

Equating the two expressions for $I$:

Dividing both sides by $V$ (since $V\mathrm{\ne }0$):

For $n$ resistors connected in parallel, the equivalent resistance is given by:

Special Case: Two Resistors in Parallel If only two resistors, $R_1$ and $R_2$, are connected in parallel, their equivalent resistance $R_p$ can be found by:

Combining the fractions on the right side:

Taking the reciprocal of both sides gives the product-over-sum rule:

Implication: The equivalent resistance in a parallel combination is always smaller than the smallest individual resistance. This is because providing more paths for the current to flow reduces the overall opposition.

Example Application: In household electrical wiring, appliances are connected in parallel. This ensures that each appliance receives the full mains voltage and can be operated independently. If one appliance is switched off or develops a fault, the others continue to function. The wiring in your home is a parallel circuit. Also, in a multi-speed electric fan, different resistances are connected in parallel with the motor winding, allowing for variable speeds by controlling the current through the motor.

Remember

In parallel, voltage is constant, and current divides. The reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances, and the equivalent resistance is always less than the smallest individual resistance.

Exam-smart

If asked to achieve the lowest possible total resistance, connect resistors in parallel. If a problem shows multiple components connected between the same two points, they are in parallel. A common mistake is to forget to take the reciprocal to find $R_p$ after calculating $\frac{1}{R_p}$.

Resistors in Parallel

When multiple resistors are connected across the same two points in an electric circuit, they are said to be in parallel. This means that the potential difference across each individual resistor is the same.

How Parallel Resistors Work

Imagine water flowing through pipes. In a parallel connection, the main water flow splits and goes through several separate pipes simultaneously. Each pipe offers some resistance to the flow. The total flow is the sum of the flows through each individual pipe.

Similarly, when resistors are connected in parallel, the electric current from the source splits and flows through each resistor. The total current entering the parallel combination is the sum of the currents flowing through each individual resistor.

Activity 11.6 Explained: This activity demonstrates that when resistors are connected in parallel:

• The potential difference across the entire parallel combination is the same as the potential difference across each individual resistor.
• The total current flowing from the source is the sum of the currents flowing through each individual resistor.

Derivation of Equivalent Resistance in Parallel:

Let $R_1,R_2,R_3,\dots$ be the resistances of individual resistors connected in parallel. Let $V$ be the potential difference across the parallel combination. This $V$ is the same for all individual resistors. Let $I$ be the total current flowing from the source. Let $I_1,I_2,I_3,\dots$ be the currents flowing through resistors $R_1,R_2,R_3,\dots$ respectively.

According to Kirchhoff's current law, the total current is the sum of the currents in each branch:

Using Ohm's Law ($V=IR$), we can express the current through each resistor as:

And so on.

Substituting these into the equation for total current:

If we consider a single equivalent resistor $R_p$ that can replace the entire parallel combination and maintain the same total current $I$ for the same potential difference $V$, then by Ohm's Law:

Equating the two expressions for $I$:

Dividing by $V$ (since $V\mathrm{\ne }0$):

This equation shows that the reciprocal of the equivalent resistance of a parallel combination is equal to the sum of the reciprocals of the individual resistances.

Key Properties of Resistors in Parallel:

• The potential difference across each resistor is the same.
• The total current is the sum of the currents through each resistor.
• The equivalent resistance is always less than the smallest individual resistance in the combination. This is because adding more parallel paths provides more routes for current to flow, thus reducing the overall opposition.

Comparison: Resistors in Series vs. Parallel

Practical Applications and Advantages of Parallel Connections:

• Household Wiring: Most electrical appliances in homes are connected in parallel. This is crucial because:
• Each appliance receives the full mains voltage (e.g., 220 V), allowing them to operate as designed.
• If one appliance fails (e.g., a bulb burns out), it does not affect the operation of other appliances.
• Different appliances require different amounts of current; parallel connections allow them to draw only the current they need.
• Avoiding High Current: In series circuits, the current is the same everywhere. If you connect devices with very different current requirements (like a lamp and a heater) in series, the device requiring less current might not work, or the entire circuit might draw too much current if one device demands it. Parallel connections solve this by allowing each device to draw its own required current independently.

Example 11.8 Explained: This example shows how to calculate the current through each resistor and the total current and resistance when resistors are connected in parallel. It emphasizes that the potential difference across each resistor is the same as the battery voltage.

Example 11.9 Explained: This example illustrates a circuit with a combination of series and parallel resistors. It demonstrates the strategy of finding equivalent resistances for parallel groups first, then combining them with series components.

Remember

In a parallel circuit, the equivalent resistance is always less than the smallest individual resistance. This is a quick way to check your calculations.

Exam-smart

Case studies often describe household wiring or complex circuits. You'll be given component values and a source voltage, and asked to find total current, voltage across a specific component, or equivalent resistance. Recognizing parallel connections is key to applying the $\frac{1}{R_{eq}}=\sum \frac{1}{R_i}$ formula and understanding that voltage is constant across all parallel branches.

Heating Effect of Electric Current

The flow of electric current through a conductor generates heat. This phenomenon is known as the heating effect of electric current. It occurs because as electrons move through the conductor, they collide with the atoms of the material, transferring energy and causing the atoms to vibrate more vigorously. This increased vibration is observed as heat.

Consider a current $I$ flowing through a resistor of resistance $R$ for a time $t$. The potential difference across the resistor is $V$. The work done ($W$) in moving a charge ($Q$) through a potential difference ($V$) is given by:

The power ($P$) supplied to the circuit by the source is the rate at which energy is supplied:

Since current $I=\frac{Q}{t}$, we can write power as:

The total energy ($E$) supplied by the source in time $t$ is:

This energy is dissipated as heat in the resistor. Therefore, the amount of heat ($H$) produced in time $t$ is:

Applying Ohm's Law ($V=IR$), we can express heat in terms of current and resistance:

This is known as Joule's Law of Heating.

Joule's Law of Heating implies that the heat produced in a resistor is:

• Directly proportional to the square of the current ($H\propto I^2$) for a given resistance and time.
• Directly proportional to the resistance ($H\propto R$) for a given current and time.
• Directly proportional to the time ($H\propto t$) for a given current and resistance.

This effect is utilized in many electrical appliances where heat generation is the desired outcome.

Practical Applications of Heating Effect of Electric Current

The heating effect of electric current, while sometimes an unwanted consequence, is ingeniously used in various domestic and industrial applications.

• Electric Heating Appliances: Devices like electric heaters, electric iron (toaster), electric ovens, and electric kettles function based on this principle. A heating element, typically made of a wire with high resistance (like nichrome alloy), is connected to the electric supply. The high resistance causes significant heat generation when current flows through it, which is then used for the intended purpose (heating a room, toasting bread, cooking food, boiling water).

• Electric Bulb (Incandescent Lamp): In an incandescent bulb, an electric current passes through a thin filament made of a material with a very high melting point, such as tungsten (melting point 3380°C). The filament gets extremely hot and emits light. To prolong its life, the filament is enclosed in a glass bulb filled with inert gases like nitrogen and argon, which prevent oxidation. Most of the electrical energy is converted into heat, with a small fraction converted into light.

• Electric Fuse: An electric fuse is a crucial safety device that protects electrical circuits and appliances from damage due to overcurrent.

• It consists of a short wire made of a metal or alloy with a low melting point (e.g., alloys of lead, tin, copper, aluminium).

• The fuse wire is connected in series with the appliance or circuit it protects.

• When an abnormally high current flows through the circuit, the fuse wire heats up rapidly due to the $I^{2}R$ effect.

• If the current exceeds a specific safe limit (the fuse rating), the fuse wire melts, breaking the circuit and stopping the flow of current. This prevents damage to the appliance and potential fire hazards.

• Fuses are rated in amperes (A), such as 1A, 2A, 3A, 5A, 10A, etc., indicating the maximum current they can safely handle. For an appliance consuming 1 kW at 220 V, the current is approximately 4.54 A, so a 5 A fuse would be appropriate.

Remember

The melting point of the fuse wire is critical. It must be low enough to melt and break the circuit before the appliance overheats, but not so low that it melts during normal operation.

Exam-smart

A question might provide the power consumed by an appliance and its operating voltage and ask you to determine the appropriate fuse rating. You would first calculate the current using $I=P\mathrm{/}V$ and then select the next standard fuse rating above this calculated current. For example, if the calculated current is 4.54 A, choose a 5 A fuse.

Electric Power

Electric power is defined as the rate at which electric energy is dissipated or consumed in an electric circuit. It can also be understood as the rate of doing work by an electric current.

The formula for electric power $(P)$ can be derived from the concepts of potential difference $(V)$, current $(I)$, and resistance $(R)$:

• Since energy is power multiplied by time $(E=P\times t)$, and from the definition of potential difference, $V=W\mathrm{/}q$, so $W=Vq$.
• Current is the rate of flow of charge, $I=q\mathrm{/}t$, so $q=It$.
• Substituting $q$ in the work equation, $W=V(It)$.
• Power is the rate of doing work, $P=W\mathrm{/}t=VIt\mathrm{/}t=VI$.

The SI unit of electric power is the watt (W). 1 watt is the power consumed by a device when 1 ampere of current flows through it at a potential difference of 1 volt.

The watt is a small unit, so in practice, the kilowatt (kW) is commonly used, where:

Formulas for Electric Power

We can express electric power in terms of resistance using Ohm's Law ($V=IR$).

1. Power in terms of Voltage and Current:

This formula is used when both the voltage across a component and the current through it are known.

2. Power in terms of Current and Resistance: Substitute $V=IR$ into $P=VI$:

This formula is useful when the current through a resistor and its resistance are known. It highlights that power dissipation is proportional to the square of the current and the resistance.

3. Power in terms of Voltage and Resistance: Substitute $I=V\mathrm{/}R$ into $P=VI$:

This formula is useful when the voltage across a resistor and its resistance are known. It shows that power dissipation is proportional to the square of the voltage.

Electric Energy and its Units

Electric energy is the total amount of work done or energy consumed by an electric device over a period of time. It is the product of power and time:

• The SI unit of electrical energy is the joule (J).
• In practice, the commercial unit of electric energy is the kilowatt hour (kW h), often called a 'unit'. This is the unit for which electricity bills are calculated.

Conversion between kilowatt hour and joule:

Applications and Examples

• Electric Bulbs: The power rating of a bulb (e.g., 100 W, 220 V) tells us the rate at which it consumes energy and dissipates it as light and heat when operated at its rated voltage. A higher power rating means it consumes more energy per second and is brighter.
• Electric Appliances: Appliances like electric irons, heaters, and refrigerators have power ratings that indicate their energy consumption rate. This is crucial for calculating electricity bills and selecting appropriate fuses. For example, an electric iron consuming 1 kW at 220 V draws a current of $I=P\mathrm{/}V=1000\text{ W}\mathrm{/}220\text{ V}\approx 4.54\text{ A}$. A fuse rated slightly higher than this, like 5 A, would be suitable.
• Energy Consumption Calculation: The cost of operating electrical appliances depends on their power rating, the duration of use, and the cost per unit of energy.

Example 1: An electric bulb is connected to a 220 V generator and draws a current of 0.50 A. What is its power?

• Using $P=VI$:

Example 2: An electric refrigerator is rated 400 W and operates 8 hours/day. Calculate the cost of energy for 30 days if the rate is Rs 3.00 per kW h.

• Energy consumed per day = Power $\times$ Time = $400\text{ W}\times 8\text{ h}=3200\text{ W h}=3.2\text{ kW h}$
• Total energy consumed in 30 days = $3.2\text{ kW h/day}\times 30\text{ days}=96\text{ kW h}$
• Total cost = Total energy $\times$ Cost per unit = $96\text{ kW h}\times \text{Rs }3.00\mathrm{/}\text{kW h}=\text{Rs }288.00$

Remember

Electrons are not consumed in an electric circuit; energy is used to move them. We pay for the electrical energy consumed, not the electrons themselves.

Exam-smart

Questions might provide voltage and current, or voltage and resistance, or current and resistance, and ask for power. They may also ask to calculate energy consumed and its cost, requiring you to convert power (W) to energy (kW h) by multiplying by time in hours. Pay attention to units.