Question

A charge Q is placed at the centre of a square. If electric field intensity due to the charge at the corners of the square is E₁ and the intensity at the mid point of the side of square is E₂, then the ratio of E₁/E₂ will be

• A. $\frac{1}{2\sqrt{2}}$
• B. $\sqrt{2}$
• C. $\frac{1}{2}$
• D. 2

Answer 1

Answer: (C)

The ratio E₁/E₂ is $\frac{1}{2}$.

Answer 2

Let the side length of the square be $a$. The charge $Q$ is placed at the center of the square.

The distance from the center of the square to any corner ($r_1$) is half the length of the diagonal. The diagonal of a square with side length $a$ is $\sqrt{a^2 + a^2} = a\sqrt{2}$.
So, $r_1 = \frac{a\sqrt{2}}{2} = \frac{a}{\sqrt{2}}$.

The electric field intensity at a corner is $E_1$. Using the formula for the electric field due to a point charge $E = \frac{kQ}{r^2}$, where $k = \frac{1}{4\pi\epsilon_0}$:
$E_1 = \frac{kQ}{r_1^2} = \frac{kQ}{(a/\sqrt{2})^2} = \frac{kQ}{a^2/2} = \frac{2kQ}{a^2}$.

The distance from the center of the square to the midpoint of any side ($r_2$) is half the side length.
So, $r_2 = \frac{a}{2}$.

The electric field intensity at the midpoint of a side is $E_2$:
$E_2 = \frac{kQ}{r_2^2} = \frac{kQ}{(a/2)^2} = \frac{kQ}{a^2/4} = \frac{4kQ}{a^2}$.

The ratio of $E_1$ to $E_2$ is:
$\frac{E_1}{E_2} = \frac{2kQ/a^2}{4kQ/a^2} = \frac{2}{4} = \frac{1}{2}$.

The ratio $E_1/E_2$ is $\frac{1}{2}$.