Question

Test charge q moves along a path P -> Q -> R -> S in a uniform electric field region directed along +x-axis. The coordinates of the points are as follows: 

P (a, b, 0), Q (2a, 0, 0), R (a, -b, 0) and S (0, 0, 0).

 

(a) Identify the plane of motion of the test charge. 

(b) Determine the work done by the electric field as the test charge moves from P to S. Use diagram, if necessary.

Answer 1

(a) As the z-coordinate of each of the points is zero, the plane of motion from P to Q to R to S is in x-y plane.

(b) Since E is conservative, the work done is path independent, so replace the path P -> Q -> R -> S with a simpler path as P -> T -> S. 

Work done along P -> T = 0, as the path is perpendicular to the direction of E. 

Work done along T -> S = -qEa 

So total work done = 0 - qEa = -qEa 

OR 

Since E is conservative, the work done is path independent, so replace the path P -> Q -> R -> S with a simpler path P -> S.

W = qE. PS. cos (90 + θ)

Here, (90 + θ) is the angle between the electric field E and displacement vector PS. θ is the ∠SPT. 

\(w = -qE\sqrt{a^2 + b^2}sin\ \theta\)

\(=-qE\sqrt{a^2 + b^2}\frac{a}{\sqrt{a^2 + b^2}}\)

\(=-qEa\)