Conducting Strips - Magnetic Field and Circuit Analysis · RMPH 2018 #3
Physics · electromagnetism
Problem
For this task you are asked to determine a number of physical quantities for systems that contain one or two conducting strips which have an electric current flowing through them. These systems are placed in air (μ_air = μ₀).
Answer the following questions and write your results as a function of the variables specified below.
Through a conducting strip of width b and length D, D≫b there is an electric current I flowing along its length. Assume that close to the strip the magnetic field lines are rectangles which surround the strip and are in planes orthogonal to it.
3.a. Find an expression which relates the induced magnetic field close to the strip to the intensity of electric current I.
We now add a second strip with the same length width D and width b and place it parallel, right above the first strip, at a distance a, a≪b≪D. The strips now both have a current I flowing along their length, but in opposite directions.
3.b. Find an expression which relates the induced magnetic field, B, between the two strips to the electric current I.
3.c. Determine the inductance L of the two strip ensemble.
The two strips have negligible electric resistance and are incorporated in the circuit presented in the figure on the right. The capacitance of the whole ensemble is also negligible. The right side of the strips are connected by a wire with negligible resistivity while the sides on the left are connected to an ideal source, generating a voltage V. Neglect all effects due to the finite speed propagation of the electric field.
3.d. Find the time dependence of the electric current I from the circuit characterized only by the inductance L, found in 3.c. Assume the initial moment (t=0) is the moment at which the circuit is closed.
3.e. Denote the distance between a point on one of the strips and the right side of the strips by x. Determine the potential between this point and the point placed identically on the other strip as a function of x.
In an infinitesimal time interval dt the energy provided by the source is given by dW = VIdt. This amount of energy is partially stored in the magnetic field to the section x; the remaining energy "flows" from section to shorted ends of strips.
3.f. Find the expression of energy flux dW(x)/dt as a function of the distance x between a point on the strip and the right end of the strip.