Question

Stress-Energy Tensor The electromagnetic stress-energy tensor is Ա Sx/c Sy/c Sz/c\ -0xx Sx/c –0ry -Oxz U S/c Τμν Sle) = S/c -ij/ Sy/c -Oyx -Oyy -oyz Sz/c - Ozx -o zy -Ozz po where u = €2E² + B2 is the energy density, S = Ex B is the Poynting vector, and 2μο ơij = €0EiEj + 1 B; B¡ – udij is the (3 × 3) Maxwell stress tensor. (Note here we use σij for the stress tensor instead of the usual Tij to avoid confusion with the symbol T.) (a) Using tensor notation, write conservation of field energy in vacuum in terms of a subset of the components of T (b) Using tensor notation, write conservation of field momentum in vacuum in terms of a subset of the components of TH. (c) Combine both conservation laws into a single 4-vector equation. (d) Consider the case of a capacitor at rest in frame S with electric field E = Eo2. Calculate Tin frame S. Now find the Poynting vector in frame 5 (moving at +v along the x-axis) by explicitly transforming the stress-energy tensor to find TV.

Answer 1

(a) Conservation of field energy in vaccum can be written in tensor notation as:

∂(uγ^0) + ∂(Siγ^i) = 0

Here, u represents the energy density, γ^0 is the unit vector along the time direction, Si represents the components of the Poynting vector, and γ^i represents the unit vectors along the spatial directions.

(b) Conservation of field momentum in vacuum can be written in tensor notation as:

∂(Tμ^0νγ^0) + ∂(Tμ^iνγ^i) = 0

Here, Tμ^ν represents the components of the stress-energy tensor, γ^0 is the unit vector along the time direction, and γ^i represents the unit vectors along the spatial directions.

(c) Combining both conservation laws into a single 4-vector equation, we have:

∂(Tμ^νγ^ν) = 0

This equation represents the conservation of the stress-energy tensor in vacuum.

(d) To calculate the stress-energy tensor T in frame S, for a capacitor at rest with electric field E = E₀^2, we need to substitute the values into the tensor components of the stress-energy tensor formula:

Tμ^ν = (1/c²)(E²δμ^ν - EμEν) + (1/µ₀c²)(B²δμ^ν - BμBν)

In frame S, the velocity v = 0. The electric field E₀ is given, so we can calculate the tensor components T⁰⁰, T⁰ⁱ, and Tⁱʲ for this frame.

To find the Poynting vector in frame S', moving at +v along the x-axis, we need to transform the stress-energy tensor T to find T'. The transformation of the stress-energy tensor is given by:

T'^μ^ν = Λ^μ_α Λ^ν_β T^α^β

Where Λ is the Lorentz transformation matrix. In this case, the Lorentz transformation is along the x-axis with velocity v. You can perform the appropriate Lorentz transformation to obtain the components of T' and then calculate the Poynting vector in frame S'.