Question

A point electric dipole is at the origin of coordinates with its dipole moment along the positive Z-axis. Consider a circular disc of radius R with its centre at z = L and its plane perpendicular to the Z-axis. The modulus of the electric flux due to the dipole through the disc is maximum when R is equal to

• A. infinity
• B. $\sqrt{2}$ L
• C. L
• D. L/$\sqrt{2}$

Answer 1

Answer: (B)

$\sqrt{2}$ L

Answer 2

To maximize the flux, we need to find the value of R for which the derivative of the flux with respect to R is zero. The flux through the disc is given by

$$\Phi = \frac{p}{2\epsilon_0} \frac{R^2}{(R^2+L^2)^{3/2}}.$$

Differentiating with respect to $R$ and equating to zero:

$$f'(R)=\frac{2R(R^2+L^2)^{3/2} - 3R^3(R^2+L^2)^{1/2}}{(R^2+L^2)^3} = 0.$$

This simplifies to:

$$2R(R^2+L^2) - 3R^3 = 0 \quad \Longrightarrow \quad 2(R^2+L^2) - 3R^2 = 0.$$

Solving,

$$2L^2 - R^2 = 0 \quad \Longrightarrow \quad R^2 = 2L^2 \quad \Longrightarrow \quad R=\sqrt{2}\, L.$$