Question

Assume a uniform electric field E= Ek exists in some region of space that contains no magnetic field. Let the magnetic vector potential A(r,t) be zero. Write down an electrostatic potential Φ(r,t) for this field. Define a gauge transformation, so that Φ (r,t)=0. What is the corresponding vector potential A ′ ?

Answer 1

1) The electrostatic potential Φ(r,t) in a uniform electric field with no magnetic field is given by Φ(r,t) = -E · r.

2) The gauge transformation Φ(r,t) = 0 implies a shift in the electrostatic potential, and the corresponding vector potential A′ is A′ = (E × r) - ∂Λ/∂t, where Λ is an arbitrary scalar function.

Step-by-step explanation:

1) In a region with a uniform electric field E and no magnetic field, the electrostatic potential Φ(r,t) is given by Φ(r,t) = -E · r, where r is the position vector. This potential satisfies the electrostatic condition.

2) The gauge transformation Φ(r,t) = 0 implies that we can choose a scalar function Λ, such that Φ' = Φ - ∂Λ/∂t = 0. This leads to the corresponding vector potential A′ = (E × r) - ∂Λ/∂t, where × denotes the cross product.