G What is the magnetic potential energy stored in a cylindrical volume of height hcylinhcylin = 50 and radius RcylinRcylin = 24 that symmetrically surrounds an infinitely long wire that has radius RwireRwire = 2.1 and carries current III = 4.9 AA ? The volume in which the energy should be calculated does not contain the wire.

Explanation:

The energy density associated to a magnetic field is:

(1)

where

B is the strength of the magnetic field

is the vacuum permeability

The magnetic field produced by a current-carrying wire is

where

I is the current in the wire

r is the distance from the wire at which the field is calculated

Substituting into (1),

(2)

Since this is the energy density, the total energy stored in a certain element of volume will be

(3)

Here the field strength changes as we move farther from the wire radially, so we can write dV as

where

h is the height of the cylinder

r is the distance from the wire

So eq(3) becomes:

Now we have to integrate this expression to find the total energy stored in the cylindrical volume. We have:

h = 50 mm = 0.050 m is the height of the cylinder

I = 4.9 A is the current in the wire

is the internal radius of the cylinder (the radius of the wire)

is the external radius of the cylinder

So,

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