Set 53 Problem number 1

Problem

Derive an expression for power required to maintain the current in a circuit with resistance R and a voltage V.

Use the expression to determine the ratio of power before and after a drop in resistance, if the resistance drops to half its original value while voltage remains constant.

If the voltage across a circuit of constant resistance is doubled, what is the ratio of the power after the increase to the power before the increase?

Solution

Power is the product of current I (measured in C/s) and voltage V (measured in J/C, so that current * voltage is in J/s). 

• From R and V we can determine I, which we then combine with V to obtain power. 

The current resulting from voltage V and resistance R is I = V / R. 

• The resulting power is therefore V I = V (V/R) = V^2 / R. 

If the resistance was cut in half, the current would double but the voltage would remain the same. That is, there would be twice as many Coulombs/second but no more Joules/Coulomb.

• The power (the number of Joules per second) would therefore double.
• This is consistent with  the presence of R in the denominator of the expression V^2 / R for power.  

For a given resistance, we can see that the current is proportional to the voltage, and that the power for a given current is also proportional to the voltage, so that voltage occurs twice as a factor in the power.

• In more concrete terms, doubling the voltage doubles the number of Coulombs per second, and also doubles the power associated with each Coulomb, so that four times the power is required to maintain the current, and four times the energy is produced by the circuit.
• This is consistent with the presence of V^2 in the expression V^2 / R for power.

Cutting the voltage in half will cut both current and voltage in half, resulting in 1/4 of the original power. This is consistent with the presence of V^2 in the expression for power.

Generalized Solution

A current I flowing across a voltage V results in power P = I V.

• Therefore current I = V / R will be associated with power P = (V / R) V = V^2 / R. 

Explanation in terms of Figure(s), Extension

The figure below charts the relationships among voltage, resistance, current and power.

• The 'blue' triangle shows how voltage, current and resistance are related.

• Greater voltage or less resistance implies greater current, expressed as I = V / R; resistance is the ratio of voltage to current, expressed as R = V / I; a current I through a resistance R requires a greater voltage drop for a greater current and for a greater resistance, expressed as V = I * R.

• The 'green' triangle shows how power is the product of current and voltage

• Current is measured in C / s, voltage in J / C, so the product of current and voltage is the number of J / s, or watts, of power.

The relationship P = I * V can be combined with either I = V / R or with V = I * R to yield either P = I * (I * R) = I^2 * R or P = (V / R ) * V = V^2 / R.

Figure(s)