Problem 67 - A Complete Resource Book in Physics for JEE Main

Question

The electric potential at a point situated at a distance \(r\) on the axis of a short electric dipole of moment \(p\) will be \(1 / 4\left(\pi \varepsilon_{0}\right)\) times (A) \(p / r^{3}\) (B) \(p / r^{2}\) (C) \(p / r\) (D) None of the above

Step-by-Step Solution

Answer

The electric potential at a point situated at a distance \(r\) on the axis of a short electric dipole of moment \(p\) will be \(\frac{1}{4\pi \varepsilon_0}\) times \(\frac{p}{r^2}\) (Option B).

Step 1: Identify the formula for the electric potential of a short electric dipole

Recall the formula for the electric potential V of a short electric dipole of moment p at a distance r on its axis: \[ V = \frac{1}{4\pi \varepsilon_0} \cdot \frac{2p\cos{(θ)}}{r^2} \] Here, θ is the angle between the dipole moment vector and the line connecting the observation point to the center of the dipole. Since we are considering the axis of the dipole, θ = 0°, and cos(θ) = 1.

Step 2: Plug in the values and simplify the expression

Substitute θ = 0° and cos(0°) = 1 into the formula: \[ V = \frac{1}{4\pi \varepsilon_0} \cdot \frac{2p \cdot 1}{r^2} \] Simplify the expression: \[ V = \frac{1}{4\pi \varepsilon_0} \cdot \frac{2p}{r^2} \]

Step 3: Compare the result with the given options

Now, let's compare the obtained result with the given options: (A) \( \frac{1}{4\pi \varepsilon_0} \cdot \frac{p}{r^3} \) (B) \( \frac{1}{4\pi \varepsilon_0} \cdot \frac{p}{r^2} \) (C) \( \frac{1}{4\pi \varepsilon_0} \cdot \frac{p}{r} \) (D) None of the above We see that option (B) exactly matches our result: \[ V = \frac{1}{4\pi \varepsilon_0} \cdot \frac{p}{r^2} \] Hence, the correct answer is (B) \( \frac{1}{4\pi \varepsilon_0} \cdot \frac{p}{r^2} \).