Combining electric potentials: just add them

Electric Fields & Potential — AQA A-Level Physics

Electric potentialThe work doneEnergy transferred when a force moves an object. In electrical circuits, W = QV (chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). times potential difference). per unit positive chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). in bringing a small test chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). from infinity to that point. is a scalar. To find the potential at a point due to multiple charges, simply add the individual potentials.
$V_total = Q1/(4\pi\varepsilon0 r1) + Q2/(4\pi\varepsilon0 r2) + ...$
No vector addition is needed -- unlike field strength, which is a vector.

Worked Example

A Van de Graaff generator has a spherical dome of radius 15 cm charged to a potential of 240 kV. Calculate (a) the charge stored and (b) the potential at 30 cm from the surface.

Show Solution

Part (a): rearrange for Q

From $V = \frac{Q}{4\pi\varepsilon_0 r}$:

$$Q = 4\pi\varepsilon_0 r V$$

Substitute for part (a)

$$Q = (4\pi \times 8.85 \times 10^{-12}) \times 0.15 \times (240 \times 10^{3})$$ $$= 4.0 \times 10^{-6} \text{ C} = 4.0 \; \mu\text{C}$$

Part (b): distance from centre

Distance from $centre = radius + 30 cm = 15 + 30 = 45 cm = 0.45 m$

Substitute for part (b)

$$V = \frac{4.0 \times 10^{-6}}{4\pi \times (8.85 \times 10^{-12}) \times 0.45} = 80 \text{ kV (2 s.f.)}$$

Answer

(a) $Q = 4.0 \; \mu\text{C}$ (b) $V = 80$ kV