Electric Potential Due to a Point Charge: MDCAT Physics notes
Electric Potential Due to a Point Charge for MDCAT: V = kq/r, scalar addition of potentials, V and E at midpoints, and V versus E ratio problems.
The formula
The potential at a distance $r$ from a point charge $q$ is the work done per unit positive charge in bringing it from infinity to that point:
$$V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r} = \frac{kq}{r}$$
- $V$ is positive near a positive charge and negative near a negative charge.
- $V \propto 1/r$: doubling the distance halves the potential.
- $V = 0$ at infinity. Potential measured from this zero is called absolute potential.
Example: potential 1 m from a $+3\ \mu$C charge is $V = 9\times10^9\times3\times10^{-6}/1 = 2.7\times10^4$ V.
Comparing $V$ and $E$ for a point charge
| Quantity | Formula | Depends on $r$ as | Nature |
|---|---|---|---|
| Field | $kq/r^2$ | $1/r^2$ | Vector |
| Potential | $kq/r$ | $1/r$ | Scalar |
Also $E = V/r$ for a point charge. To link the two in ratio questions, first find how $r$ changes from the given quantity, then apply it to the other.
Example: at a point the field is $E$ and potential $V$. Where the field is $E/9$, the distance is $3r$, so the potential is $V/3$.
Several charges: add as scalars
Because potential is a scalar, the total potential at a point is the algebraic sum of the potentials of all charges, taking signs into account. Fields, being vectors, must be added with direction.
Midpoint between two charges
| Charges | Potential at midpoint | Field at midpoint |
|---|---|---|
| $+q$ and $+q$ (equal like) | $2kq/d \neq 0$ (adds) | Zero (equal and opposite) |
| $+q$ and $-q$ (equal unlike) | Zero (cancels) | Not zero (adds, points to $-q$) |
Here $d$ is the distance from each charge to the midpoint. These two results are opposite of each other and are frequently tested together.
Worked example
Charges $+2\ \mu$C and $-6\ \mu$C are 4 m apart. Potential at the midpoint (2 m from each):
$$V = \frac{9\times10^9(2\times10^{-6})}{2} + \frac{9\times10^9(-6\times10^{-6})}{2} = 9000 - 27000 = -18000\ \text{V}$$
Equipotentials
Around a point charge, surfaces of constant potential are spheres centred on the charge. No work is done moving a charge along an equipotential, and field lines cross equipotentials at right angles.
Key formulas
- $V = kq/r$
- $E = kq/r^2 = V/r$
- $V_{\text{total}} = \sum kq_i/r_i$ (with signs)
Common MDCAT traps
- Using $1/r^2$ for potential. Potential falls as $1/r$.
- Midpoint of equal like charges: $E = 0$ but $V \neq 0$.
- Midpoint of equal unlike charges: $V = 0$ but $E \neq 0$.
- Forgetting the micro prefix: $1\ \mu$C $= 10^{-6}$ C.
Quick revision
- Potential is a scalar; add potentials with signs.
- Doubling $r$ halves $V$ and quarters $E$.
- Absolute potential is referred to zero at infinity.
- Equipotentials around a point charge are concentric spheres.