Question

A constant voltage is applied across a circuit. If the resistance in the circuit is doubled, what is the effect on the power dissipated by the circuit?

	The power dissipated is reduced by a factor of 4.
	The power dissipated is reduced by a factor of 2.
	The power dissipated is doubled.
	The power dissipated remains constant.
	The power dissipated is quadrupled.

If the resistance in a circuit connected to a constant current is halved, how is the voltage in the circuit affected?

	The voltage is reduced by a factor of 4.
	The voltage is doubled.
	The voltage is reduced by a factor of 2.
	The voltage is quadrupled.
	The voltage remains constant.

Answer 1

Concept and Reason

The main concept used is ohm’s law and power relation with the current and resistance.

Initially, use the power relation to calculate the effect on the power dissipated by the circuit. Finally use the ohm’s law to calculate how voltage in the circuit affected.

Fundamental

The power dissipation across resistors is given by,

$P = {I^2}R$

Here,$P$is the power,$I$is the current and $R$is the resistance the circuit.

From the Ohm’s law, the expression of the voltage is as follows,

$V = IR$

Here,$V$is the voltage.

Calculate the power dissipated by resistor if the resistance of the circuit is double.

The expression of the power in terms of the voltage is as follows,

$P = \frac{{{V^2}}}{R}$

From the above expression of the power it is clear that if the voltage of the circuit is constant then the power expression of the circuit is inversely propositional to the resistance of the circuit.

Therefore, the expression of the power is as follows,

$\frac{{{P_1}}}{{{P_2}}} = \frac{{{R_2}}}{{{R_1}}}$

Substitute $2R$for${R_2}$and $R$for${R_1}$in the above expression of power,

$\begin{array}{c}\\\frac{{{P_1}}}{{{P_2}}} = \frac{{2R}}{R}\\\\{P_2} = \frac{{{P_1}}}{2}\\\end{array}$

Calculate the voltage in the circuit affected if the resistance of the circuit is half.

From Ohm’s law, the expression of the voltage is as follows,

$V = IR$

From the above expression of the voltage, voltage is directly proportional to the resistance of the circuit if the current is constant.

Therefore, the above expression of the voltage is as follows,

$\frac{{{V_1}}}{{{V_2}}} = \frac{{{R_1}}}{{{R_2}}}$

Rearrange the above expression in terms of the voltage ${V_2}$,

${V_2} = \frac{{{V_1}{R_2}}}{{{R_1}}}$

Substitute $\frac{R}{2}$for${R_2}$and $R$for${R_1}$in the above expression of the voltage ${V_2}$

$\begin{array}{c}\\{V_2} = \frac{{{V_1}R}}{{2R}}\\\\ = \frac{{{V_1}}}{2}\\\end{array}$

Ans:

The power dissipated is by the resistor is half.

If the resistance of the circuit is half, then voltage is half.