Potential difference is work done per unit charge

Potential difference (V) is the work done (W) per unit charge (Q):

$\text{V}=\frac{\text{W}}{\text{Q}}$

Where:

V = potential difference (V)

W = work done (J)

Q = Charge (C)

A potential difference of one volt means one joule of work is transferred for each coulomb of charge moving through it:

$1\text{ V}=\frac{1\text{ J}}{1\text{ C}}$

Worked example 1 - Calculating potential difference

Calculate the potential difference across a component if 50 joules of work is done to move a charge of 10 coulombs.

Step 1: Formula

$\text{V = }\frac{\text{W}}{\text{Q}}$

Step 2: Substitution and correct evaluation

$\text{V = }\frac{50}{10}=5\text{ V}$

Relating potential difference to kinetic energy

When a charge is accelerated due to a potential difference, it gains kinetic energy. For an electron with a charge of -e, the work done (W) can be equated to its kinetic energy:

$\text{W = V e = }\frac{1}{2}{\text{ m v}}^2$

Where:

W = work done (J)

V = potential difference (V)

e = charge of an electron (1.6 $\times {10}^{-19}$ C)

m = mass of electron ($9.11\times {10}^{-31}$ kg)

v = velocity (m s${}^{-1}$)

Worked example 2 - Kinetic energy of an electron

Calculate the speed of an electron accelerated through a potential difference of 12 volts.

Step 1: Formula

$\text{W = V e = }\frac{1}{2}{\text{m v}}^2$

Step 2: Calculate Work Done (W)

$\text{W = V }\times \text{ e}$

$\text{W = }12\times -1.6\times {10}^{-19}=-1.92\times {10}^{-18}\text{ J}$

Step 3: Calculate electron speed

$\text{v = }\sqrt{\frac{2\times W}{m}}$

$\text{v = }\sqrt{\frac{2\times -1.92\times {10}^{-18}}{9.11\times {10}^{-31}}}=2.05\times {10}^6{\text{ m s}}^{-1}$

Power is the rate of energy transfer

Power (P) measures the rate of energy transfer or rate of doing work:

$\text{P = }\frac{\text{W}}{\text{t}}$

Where:

P = power (W)

W = Work done (J)

t = time (s)

Power has a simple formula for electrical circuits:

$\text{P = V I}$

Where:

P = power (W)

V = potential difference (V)

I = current (A)

Other power equations

We know $V=IR$ from the definition of resistance $R$. Substituting this into $P=VI$ gives:

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

${\text{P = I}}^2\text{ R}$

The choice of equation depends on the quantities provided.

Electrical energy

Work done (W) is power multiplied by time:

$\text{W = P t = V I t}$

Where:

W = energy transferred (W)

P = Power (W)

t = times (s)

V = potential difference (V)

I = current (A)

Worked example 3 - Calculating energy transfer

An electric kettle draws a current of 4A when connected to the 230V mains supply. It takes 270 seconds to boil the water.

Calculate the electrical energy transferred.

Step 1: Formula

$\text{W = V I t}$

Step 2: Substitution and correct evaluation

$\text{W = 230}\times 4\times 270=\mathrm{248,400}\text{ J}$