Power Dissipated across Resistor

Power dissipation is the process of loss of power in the form of heat due to primary action. It is a naturally occurring process. All the resistors that are part of the circuit and have a voltage drop across them will dissipate power. The electrical power gets converted to heat energy, and therefore all the resistors will have a power rating. The power rating is the maximum power that can be dissipated from a resistor without burning out.

Power Dissipated in a Resistor

The equation of power is P = v x i

P = i² R [ since v = iR]

v is the voltage.

i is the current.

R is the resistance.

Maximum Power Dissipated across a Resistor

Let us consider that a cell with an emf E and an internal resistance r is connected to an external resistance R.

The value of the current is given by the formula

i = E/(r + R)

E is the emf of the cell.

r is the internal resistance.

R is the external resistance.

The power dissipated in the resistor is given by the formula

P = i² R

Substituting the value of current “i” in the above equation,

i = E/(r + R)

P = [E/(r + R)]² R …..(1)

The condition for maximum power dissipation can be calculated by differentiating power with respect to R and then equating it to zero.

Differentiating the equation (1) with respect to R,

The above equation is equated to zero to find the condition for maximum power dissipation

⇒ dP/dR = 0

⇒ (r + R)² -2(r+R)R = 0

(r + R)² = 2(r+R)R

(r² + R² +2rR) = 2rR

⇒ R² = r²

R = r

When the value of the external resistance is equal to the internal resistance, then the power will be maximum.

Substituting R = r in equation (1), we get

P = [E/(r + r)]² r

Maximum power, P = E²/4r

Maximum power is dissipated across a variable external resistance when that external resistance (R) matches the internal resistance(r), and when this happens, the value of maximum power is equal to E²/4r.

The graph shows how the power varies with external resistance R.

Power Dissipated across Resistor – Video Lesson

Solved Example

1) What is the power dissipated in a 10kΩ resistor when a current of 5mA passes through the resistor?

Answer:

Power, P = i² x R

= (5 x 10^-3)² x 10 x 10³

= 250 mW

The power dissipated in the resistor is 250 mW.