# Maximum Horizontal Force of Relativistic Point Charge

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1. Feb 14, 2019 at 2:17 AM

### QuantumDuck23

1. The problem statement, all variables and given/known data
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)

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

3. 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.

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• ###### IMG_1218.jpg
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2. Feb 14, 2019 at 12:36 PM

### TSny

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.

3. Feb 14, 2019 at 7:40 PM

### QuantumDuck23

Found the error. Thank you.