Set 52 Problem number 5

Problem

A potential difference of 10 volts is maintained across a resistance of 30 ohms. How much current results, and how much power is required to maintain this current?

Solution

A greater resistance implies a lesser current, while a greater voltage implies a greater current.

The current is seen from preceding problems to be proportional to the voltage and inversely proportional to the resistance.
We thus have I = c V / R for some proportionality constant c.

By definition of an ohm a circuit has a resistance of 1 ohm if a 1 volt potential difference results in a current of 1 ampere.

It follows immediately that when I is in amperes and V in volts, the proportionality constant c must be 1.
Therefore I = V / R, where I is in amperes, V in volts and R in ohms.

Thus the 10 volt potential difference and the 30 ohm resistance result in a current of I = 10 volts / ( 30 ohms) = .3333 volts/ohm = .3333 amperes, or .3333 C/s.

A current of .3333 C/s through a potential difference of 10 volts results in ( .3333 C/s)( 10 J/C) = 3.333 J/s = 3.333 watts.

Generalized Solution

An ohm is the unit of resistance which permits 1 ampere of current to flow in response to a 1 volt potential difference.

For a given voltage, current is inversely proportional to resistance, so we write I = V / R.
The resistance of a length of uniform wire of a given homogeneous material is directly proportional to its length and inversely proportional to its cross-sectional area.

greater length implies proportionally less potential gradient for a given voltage and hence proportionally less current; greater area implies proportionally more available charge carriers per unit of length and hence proportionally more current.

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.