A point charge travels along a straight line with a constant velocity v . Consider a small area A oriented perpendicular to the direction of motion of the charge (in the figure). Determine the displacement current through the area when its distance from the charge is x . The value of x is not large, such that the electric field at any instant is essentially given by Coulomb's law.

Step-by-step solution

The electric field at a distance x from the point charge is given by Coulomb's law: E = q 4 ₀ x^2 Since the area A is small and oriented perpendicular to the direction of motion, the electric field is uniform over the area and parallel to the area vector. The electric flux through the area A is: _E = E A = q A 4 ₀ x^2 The displacement current I_d is defined as: I_d = ₀ d _E dt Differentiating the electric flux with respect to time, we get: d _E dt = d dt ( q A 4 ₀ x^2 ) = q A 4 ₀ ( - 2 x^3 ) dx dt Since the charge