Q.Two point charges of magnitude $+q$ and $-q$ are placed at $(-d/2, 0, 0)$ and $(d/2, 0, 0)$, respectively. Find the equation of the equipotential surface where the potential is zero.

Concept: Electric Potential — The potential at a point due to a system of point charges is the scalar sum of the potentials from each charge. For a zero-potential surface, the sum is set to zero.

Step 1: Write the potential at a general point $(x, y, z)$.

The distance from $+q$ at $(-d/2, 0, 0)$ is $r_1 = \sqrt{(x + d/2)^2 + y^2 + z^2}$.

The distance from $-q$ at $(d/2, 0, 0)$ is $r_2 = \sqrt{(x - d/2)^2 + y^2 + z^2}$.

Potential:

Step 2: Set $V = 0$.

$\frac{q}{r_1} - \frac{q}{r_2} = 0 \quad\Rightarrow\quad \frac{1}{r_1} = \frac{1}{r_2} \quad\Rightarrow\quad r_1 = r_2$ …

Why this works — the idea of electric potential

Electric potential is scalar, so the total potential at any point is simply the algebraic sum of the potentials due to each charge. For a point charge, $V = \frac{kq}{r}$, where $r$ is the distance from the charge. The sign of the charge matters: positive charges give positive potential, negative charges give negative potential.

When we look for points where the total potential is zero, we are essentially finding the set of locations where the contributions from $+q$ and $-q$ exactly cancel. Since the two charges have equal magnitude but opposite sign, cancellation happens when the distances to the two charges are equal — because then $kq/r_1 + k(-q)/r_2 = kq(1/r_1 - 1/r_2) = 0$ implies $r_1 = r_2$.

So the problem reduces to: find all points $(x,y,z)$ such that the distance to $(-d/2,0,0)$ equals the distance to $(d/2,0,0)$.

Step-by-step solution

1. Write the condition for zero potential

Let $r_1$ be the distance from a point $P(x,y,z)$ to the charge $+q$ at $(-d/2,0,0)$, and $r_2$ the distance to $-q$ at $(d/2,0,0)$.

This gives $\frac{1}{r_1} = \frac{1}{r_2}$, so $r_1 = r_2$.

2. Express the distances in coordinates

Setting them equal:

3. Square both sides and simplify

Squaring eliminates the square roots:

Cancel $y^2 + z^2$ from both sides:

4. Expand and solve for $x$

Cancel $x^2$ and $\frac{d^2}{4}$:

Since $d \neq 0$, we get $x = 0$. …

Method: Finding an Equipotential Surface for a System of Point Charges

Use this method whenever asked to find the shape/equation of the surface where a system of two or more point charges produces a SPECIFIC potential value (commonly zero).

Steps

Step 1: Write the total potential as a sum over all charges

Since potential is a scalar, the total potential at a general point $(x,y,z)$ is just the algebraic sum of each charge's contribution:

where $r_i$ is the distance from the point to the $i$-th charge. Write each $r_i$ explicitly in coordinates using the given charge positions.

Step 2: Set the sum equal to the target value

Set $V(x,y,z)$ equal to the potential value specified in the problem (often $0$) and simplify. For two charges of equal and opposite magnitude, the zero-potential condition reduces to the simple statement that the two distances must be equal: $r_1 = r_2$.

Step 3: Square both sides to remove square roots and simplify …