Maxwell's Equations

Part a$)(10$ points$)$

You are given a circuit with a DC voltage source connected to two resistors in series, with ideal wires connecting all components. The resistors are arranged such that the wires are separated by a constant distance d$.$ The circuit is as follows:

Voltage source: V $=12$ V

Resistor $1$: R$1=100$ Ohm

Resistor $2$: R$2=200$ Ohm

Distance between the wires: d $=0\mathrm{.}5$ m

Using Maxwell's equations, calculate the electric field E in specific points of the circuit. Assume the wires are ideal and the magnetic field contribution is negligible.

Calculate the electric field at a point in the wire between the two resistors.

Find the magnetic field at a point near the wire, and calculate its value using the right$-$hand rule.

What is the total current in the circuit?

Hint: Use Ohm's law and the principles of Maxwell's equations for steady$-$state conditions. Use a mathematical contour for calculating the fields.

Part b$)(10$ points$)$

Consider a transmission line consisting of two parallel wires connected to the DC voltage source. The separation between the wires is d$,$ and the wires have a radius r$.$ Calculate the inductance per unit length of this transmission line, assuming the wires are in a vacuum.

Derive an equation for the inductance per unit length of this transmission line using the geometry of the system.

Find the magnetic flux density in the region between the wires.

Use the right$-$hand rule to calculate the direction of the magnetic field lines.

Given:

Wire separation d $=0\mathrm{.}5$ m$,$

Wire radius r $=0\mathrm{.}01$ m$,$

Permeability of free space mu$\_0=4$pi x $10^-7$ H$/$m$.$