Amp it up!

Electricity and Magnetism Level 4

A wire is aligned with the $z$ -axis and carrying a current, with a steady current density of $\vec{J}=(0,0,\sqrt{x^2+y^2})$ for $x^2+y^2\leq 36$ . The current produces a magnetic field $\vec{B}(x,y,z)$ ; Amp $\grave{\text{e}}$ re's circuital law tells us that $\nabla \times \vec{B}= \mu_0\vec{J}$ , where $\mu_o$ is the magnetic constant.

Find $\frac{||\vec{B}(8,6,4)||}{\mu_0}$ .

(from a recent test on vector calculus)

The answer is 7.2.

1 solution

Steven Chase
Dec 1, 2018

I prefer the integral form of Ampere's law:

$\int_C \vec{B} \cdot \vec{d \ell} = \mu_0 \, I_{enclosed}$

Since the magnetic field is circular and constant in magnitude at a given radius, this reduces to:

$2 \pi \, R \, B = \mu_0 \, I_{enclosed} \\ R = \sqrt{8^2 + 6^2} = 10 \\ B = \frac{\mu_0 \, I_{enclosed}}{20 \pi}$

Now to find the enclosed current:

$| \vec{J} | = r \\ dA = 2 \pi r \, dr \\ d I = | \vec{J} | \, d A = 2 \pi r^2 \, dr \\ 0 \leq r \leq 6 \\ I_{enclosed} = 2 \pi \, \int_0^6 r^2 \, dr = 144 \pi$

The B-field at the measurement point is therefore:

$B = \frac{\mu_0 \, 144 \pi}{20 \pi} \\ \frac{B}{\mu_0} = \frac{36}{5} = 7.2$

Yes, exactly!! Very clearly and lucidly explained! Thank you!

My vector calculus class is about Maths, not Physics, at least according to the course catalogue. Maxwell's equations are not discussed in class, and the students are not expected to know them. In this problem, they are supposed to use Stokes' Theorem to get from the given differential form of Ampere's law to the integral form:

$\int_{\partial S} \vec{B} \cdot d\vec{s}=\int_S curl\vec{B}\cdot d\vec{S}=\mu_0 \int_S \vec{J}\cdot d\vec{S}$ where $S$ is the disk $x^2+y^2\leq 100 , z=4$ . The ensuing computations are exactly those you performed, of course.

Otto Bretscher - 2 years, 6 months ago

Amp it up!

The answer is 7.2.

1 solution

0 pending reports