Understanding Energy Supplied in an Electrical Circuit

Let's break down how to find the energy supplied by a source to a resistor when a current flows through it. We are given the potential difference (\(V\)) across the resistor, the current (\(I\)) flowing through it, and the time (\(t\)) for which the current flows.

Defining Key Terms in Electrical Circuits

• Potential Difference (\(V\)): This is the work done per unit charge to move a charge between two points in an electric field. It is also called voltage. Measured in volts (V).
• Current (\(I\)): This is the rate of flow of electric charge. Measured in amperes (A).
• Resistor: An electrical component that opposes the flow of electric current. Its property is resistance (R), measured in ohms (\(\Omega\)).
• Energy: The capacity to do work. In an electrical circuit, energy is transferred from the source (like a battery or power supply) to components like resistors, where it is often dissipated as heat. Measured in joules (J).
• Power (\(P\)): The rate at which energy is transferred or converted. In electrical circuits, it's the rate at which the source supplies energy or a component dissipates energy. Measured in watts (W).

Relating Power, Voltage, and Current

The electrical power (\(P\)) supplied to a component in a circuit is directly proportional to both the potential difference (\(V\)) across it and the current (\(I\)) flowing through it. The fundamental formula for electrical power is:

\[P = V \times I\]

This formula tells us the rate at which energy is being supplied or consumed in the circuit component at any given moment.

Calculating Energy Supplied Over Time

Energy is the total amount of power supplied or consumed over a specific period of time. If power (\(P\)) is constant over a time interval (\(t\)), the total energy (\(E\)) supplied or consumed is given by the formula:

\[E = P \times t\]

This means if power is measured in watts (which is joules per second) and time in seconds, the energy will be in joules.

Deriving the Energy Supplied Formula

Now, we can combine the two formulas we have:

1. Power: \(P = V \times I\)
2. Energy: \(E = P \times t\)

Substitute the expression for power (\(P = V \times I\)) from the first formula into the second formula:

\[E = (V \times I) \times t\]

So, the energy (\(E\)) supplied by the source in time (\(t\)) when a potential difference (\(V\)) is maintained across a resistor and a current (\(I\)) flows through it is:

\[E = VIt\]

Comparing with Options

Let's compare our derived formula with the given options:

• Option 1: \(V I t^2\) (Incorrect)
• Option 2: \(V I\) (This represents power, not energy) (Incorrect)
• Option 3: \(V I t\) (Correct)
• Option 4: \(V I / t\) (Incorrect)

Our derived formula \(E = VIt\) matches the third option.

Revision Table: Electrical Energy Formulas

Additional Information: Related Electrical Concepts

The energy dissipated in a resistor is converted into heat. This phenomenon is described by Joule's Law of Heating. The heat energy (\(H\)) produced in a resistor with resistance \(R\) when a current \(I\) flows through it for time \(t\) is given by:

\[H = I^2Rt\]

Using Ohm's Law, which states \(V = IR\), we can express this energy formula in other ways:

• Substitute \(R = V/I\) into \(H = I^2Rt\): \[H = I^2 \left(\frac{V}{I}\right) t = VI t\] This shows that the heat dissipated (\(H\)) is equal to the energy supplied (\(VIt\)) by the source, assuming all supplied energy is converted to heat in the resistor.
• Substitute \(I = V/R\) into \(H = I^2Rt\): \[H = \left(\frac{V}{R}\right)^2 Rt = \frac{V^2}{R^2} Rt = \frac{V^2}{R} t\] So, \(H = \frac{V^2}{R} t\).

All three expressions for energy/heat \((VIt, I^2Rt, V^2t/R)\) are equivalent and represent the energy supplied by the source and dissipated by the resistor in time \(t\).