Induced Magnetic Fields — Question 1

Question 1

A rising axial electric field produces an azimuthal magnetic field

In a vacuum cylindrical region of radius a, $E_z=\beta t$ is uniform; its changing flux is negligible outside. No conduction current crosses the circular contours. Use $\oint\mathbf B\cdot d\mathbf l=c^{-2}d\Phi_E/dt$ in the quasistatic approximation.

1. Find Bphi inside and outside the region.

2. Find B at r=.0300 m for a=.0600 m, beta=$10^{12}$ V/(m s), and $c=3.00\times10^8$ m/s.

Question 1 - Solution

A rising axial electric field produces an azimuthal magnetic field

(a) Method and calculation

$$2\pi rB_\phi=c^{-2}\beta\pi\min(r^2,a^2).$$$$\boxed{B_\phi=\beta r/(2c^2)\ (r<a);\quad B_\phi=\beta a^2/(2c^2r)\ (r>a)}.$$

(b) Interpretation and further calculation

$$B(.0300)=10^{12}(.0300)/[2(3.00\times10^8)^2].$$$$\boxed{B=1.66667\times10^{-7}\,T\quad\text{counterclockwise from }+z}.$$

Physical check

The right-hand circulation follows the increasing electric flux, not the electric-field magnitude alone.