Question

Consider the following conclusions regarding the components of an electric field at a certain point in space given by

$E_x = -Ky$ $E_y = Kx$ $E_z = 0$

(I) The field is conservative. (II) The field is non-conservative.

(III) The lines of force are staright lines (IV) The lines of force are circles.

Of these conclusions

• A. II and IV are valid
• B. I and III are valid
• C. I and IV are valid
• D. II and III are valid

Answer 1

Answer: (A)

A

Answer 2

To determine if the electric field is conservative, we calculate its curl. If the curl is zero, the field is conservative; otherwise, it's non-conservative.

Given $E_x = -Ky$, $E_y = Kx$, $E_z = 0$, the curl is:

$\nabla \times \vec{E} = \left( \frac{\partial E_z}{\partial y} - \frac{\partial E_y}{\partial z} \right) \hat{i} + \left( \frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x} \right) \hat{j} + \left( \frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y} \right) \hat{k} = (0 - 0)\hat{i} + (0 - 0)\hat{j} + (K - (-K))\hat{k} = 2K\hat{k}$.

Since the curl is non-zero ($2K\hat{k} \neq 0$), the field is non-conservative.

To find the shape of the field lines, we use the relationship $\frac{dx}{E_x} = \frac{dy}{E_y}$.

$\frac{dx}{-Ky} = \frac{dy}{Kx} \implies x \, dx = -y \, dy$.

Integrating both sides gives $\int x \, dx = \int -y \, dy \implies \frac{x^2}{2} = -\frac{y^2}{2} + C$.

Rearranging, we get $x^2 + y^2 = 2C$, which is the equation of a circle.

Therefore, the correct conclusions are (II) The field is non-conservative and (IV) The lines of force are circles.