Maximum Horizontal Force of Relativistic Point Charge

[QuantumDuck23]

Homework Statement

A charge q1 is at rest at the origin, and a charge q2 moves with speed βc in the x-direction, along the line z = b. For what angle θ shown in the figure will the horizontal component of the force on q1 be maximum? What is θ in the β ≈ 1 and β ≈ 0 limits? (see image)

Homework Equations

Equation for the electric field of a stationary point charge: Q/(4*pi*ε*R^2)
Lorentz transformations

The Attempt at a Solution

Starting out with the equation for the electric field of a stationary point charge, I used the Lorentz transformations to transform the electric field expression to the reference frame moving with βc and multiplied that expression by z / ((γx)^2 + z^2)^(1/2) (equivalent to cos θ) to account for the horizontal component of the electric field. I took the derivative of this electric field expression to obtain the x-value for which the horizontal electric field was at a maximum, which was at x = b/(sqrt(2) * γ), and I rewrote this value in terms of sin θ (since the problem is asking for it in terms of θ). My final answer was sin θ = sqrt(2γ/(2γ + 1)), but that does not match up with the actual solution, which is sin θ = sqrt(2 / (3 - β^2), although for the limiting cases of β = 1 and β = 0 they are identical.

 

[TSny]

which was at x = b/(sqrt(2) * γ)

OK. Note that γ is not inside the square root.

and I rewrote this value in terms of sin θ (since the problem is asking for it in terms of θ). My final answer was sin θ = sqrt(2γ/(2γ + 1))

This isn't correct. I think you might have made a careless error here. Check your work.

 

[QuantumDuck23]

OK. Note that γ is not inside the square root.

This isn't correct. I think you might have made a careless error here. Check your work.

Found the error. Thank you.