An infinitely long line of charge has linear charge density ë. Find the energy density of the electric field at a point a radial distance r from the wire. (Use the following as necessary:
å0, ë, r.)
Do you how to use Gauss law to determine the electric Field E at some point r away from the line?
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html
Then Energy density is E/volume, and that works out to be 1/2 epsilonnaught*E^2
To find the energy density of the electric field at a radial distance r from the wire, we need to use the formula for energy density, which is given by:
u = (1/2) * ε0 * E^2
where u is the energy density, ε0 is the permittivity of free space, and E is the electric field.
In this case, we have an infinitely long line of charge with linear charge density λ, and we want to find the energy density at a radial distance r from the wire.
First, we need to determine the electric field E at that point. The electric field created by an infinitely long line of charge can be calculated using Coulomb's Law. The formula for the electric field at a point P located at a radial distance r from the wire is given by:
E = (λ / 2πε0r)
where λ is the linear charge density, ε0 is the permittivity of free space, and r is the radial distance from the wire.
Now that we have the electric field E, we can calculate the energy density u using the formula:
u = (1/2) * ε0 * E^2
Substituting the value of E into the formula, we get:
u = (1/2) * ε0 * ((λ / 2πε0r) )^2
Simplifying further:
u = (1/2) * ε0 * (λ^2 / (4π^2ε0^2r^2) )
Since ε0 is a constant, we can simplify the expression to:
u = (λ^2 / (8π^2ε0r^2) )
Therefore, the energy density of the electric field at a radial distance r from the wire is given by:
u = (λ^2 / (8π^2ε0r^2) )