(a) Show that the magnetic dipole moment of a magnetized...

(a) Show that the magnetic dipole moment of a magnetized body can be written $m = \int d^3 r \, r \rho^*(r)$, where $\rho^*(r) = -\nabla \cdot M(r)$ is the density of fictitious magnetic charge. (b) Let $B_1$ and $B_2$ be the magnetic fields produced by bodies with magnetizations $M_1$ and $M_2$, respectively. Use the fact that magnetization current density is as \"real\" as electric current density to show that the potential energy of interaction between the two bodies satisfies the reciprocity relation $$V_B = -\int d^3 r \, M_1 \cdot B_2 = -\int d^3 r \, M_2 \cdot B_1$.

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00:01 Question 42.

00:03 Part a asks us to show that the energy densities for electric field and magnetic field, ue and ub, are equivalent, are such that gives a hint that show that the waves energy is divided equally between the electric field and magnetic field.

00:19 So we don't look for this expression for energy densities.

00:22 We can start with the typical definition in terms of electric and magnetic fields.

00:26 We see that the energy density of the electric field is 1 half, epsilon not times energy squared, and the energy density of the magnetic field is b0, sorry, b squared over 2 mu not.

00:47 So i'm going to convert one to the other, or just show that they provide the same value.

00:52 So what i'm going to do, i'm going to use a relationship that the electric field is equivalent to the speed of light multiplied by the electric field.

01:00 I'm going to go ahead and insert that into m .e.

01:02 So i have u e is equal to one half epsilon not times c squared b squared.

01:16 Additionally, i'm going to use the relationship that epsilon not mu not is equivalent to one over c squared.

01:26 So one half epsilon knot and then c squared is then equivalent to one over epsilon not mu not times b squared.

01:40 With this, i should be able to show that my epsilon knots cancel out.

01:43 I have b squared divided by 2 mu not...

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