A length of copper wire carries a current of 10 A uniformly...

A length of copper wire carries a current of 10 A uniformly distributed through its cross section. Calculate the energy density of (a) the magnetic field and (b) the electric field at the surface of the wire. The wire diameter is $2.5 \mathrm{mm},$ and its resistance per unit length is 3.3$\Omega / \mathrm{km}$ .

A length of copper wire carries a current of 10 A uniformly
distributed through its cross section. Calculate the energy density
of (a) the magnetic field and (b) the electric field at the surface of
the wire. The wire diameter is $2.5 \mathrm{mm},$ and its resistance per unit
length is 3.3$\Omega / \mathrm{km}$ .

Key Concepts

Magnetic Field Energy Density

This concept refers to the amount of energy stored in a magnetic field per unit volume. It is expressed by the formula u_B = B²/(2??), where B is the magnetic field strength and ?? is the permeability of free space. Understanding this concept is crucial in electromagnetism and is applied to systems where currents produce magnetic fields.

Electric Field Energy Density

This concept describes the energy stored in an electric field per unit volume and is calculated by the formula u_E = ½??E², where E is the electric field strength and ?? is the permittivity of free space. It plays an important role in understanding how energy is stored in systems like capacitors and resistive circuits.

Uniform Current Distribution

Uniform current distribution means that the electric current is spread evenly across the cross-section of a conductor. This assumption simplifies the calculation of current density and the resulting magnetic field, and it is fundamental in modeling the behavior of currents in conductive materials.

Resistance per Unit Length

Resistance per unit length quantifies how the resistance of a conductor is distributed along its length. It is particularly relevant in understanding the voltage drop and internal electric fields within a conductor, which in turn are important for determining the work done by or on the charge carriers in a circuit.

Transcript

00:01 Okay, so for this problem, we have a wire carrying some current and our goal is to get the field densities for the electric and magnetic fields.

00:10 And i'll just say right off the bat that this problem took me way longer than i expected.

00:17 But that's okay.

00:18 It happens sometimes in physics.

00:20 So maybe you are kind of in front too and you're like, oh my god, why did this take so long? i forgot to half the diameter of the wire.

00:33 Okay, that was one of my issues, and then, yeah, anyway, i didn't.

00:39 It was like, why is there an electric field? so i kind of get to that as each part comes up.

00:45 After i write down the given.

00:46 So, yeah, we have the diameter is 2 .5, so the radius is 2 .5 millimeters divided by 2 .2.

00:56 And the resistance per length.

01:02 I'll just write it like that is 3 .3 oms per kilometer, which is a thousand meters.

01:11 You see my video, you know, i like to put everything in si units as soon as possible.

01:18 Okay, and so for the energy density, we want to evaluate b squared over two mu not.

01:27 And b for a wire is, um, mu not i over to pi r.

01:38 So plugging that in, oops.

01:41 Oh, let me make this a little r so that my um resistance and radius have distinct symbols.

01:53 Great.

01:53 And then squaring that...

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