Spring 2023 Homework Assignments
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A toroidal transformer is constructed using a (linear) material with permeability μ in the shape of a torus and with the dimensions given in the figure. One winding (terminals 1 and 2) has N turns and the second winding is a single wire (terminals 3 and 4) going through the torus. (Such a structure is called a "current transformer" because it is useful for measuring the current through the wire.) (a) If I1 is zero, what is the potential difference across the terminals 1 and 2 as a function of current I2? (b) Find the self inductance L1 for the winding with terminals 1 nd 2. (c) Find the mutual inductance M between the two windings. (Note that M = M12 = M21 so use the easier one to calculate.) (d) Now, assume I2 is zero and and I1 has a constant value. Find the magnetic field and show that the energy associated with the magnetic field is equal to the energy LI2/2 stored in the inductor. (Note that you will have to use the quantity uB = B⋅H/2 for the energy density inside the torus.)
Consider two line charges, parallel to the z-axis, placed on the x-axis at positions x=-a and x=a. The line charges have λ charge per unit length. (a) What is the direction and magnitude of the electric field on the y-z plane as a function of y and z? (b) Find the force per unit length acting on one of the line charges due to the field from the other line charge. (c) Integrate the appropriate component of the Maxwell stress tensor over the y-z plane to obtain the force in the previous part. The line charges are now set to motion so that they both move in the z-direction with the same constant speed v. (d) What is the direction and magnitude of the magnetic field on the y-z plane as a function of y and z? (e) Repeat part (b) for the effect of the magnetic field. (f) Repeat part (c) for the effect of the magnetic field.
An electromagnetic wave is incident perpendicularly (in the z-direction) onto a thin dielectric material with thickness a and index of refraction n=3. (The magnetic permeability of the material is equal to that of free space.) The wave structure is expected to have the form For z<0 : Incident wave: EI exp[i( kz-ωt)] Reflected wave: ER exp[i(-kz-ωt)] For 0<z<a : Wave in +z dir.: E2 exp[i( Kz-ωt)] Wave in -z dir.: E3 exp[i(-Kz-ωt)] For z>a : Transmitted wave: ET exp[i( kz-ωt)] Note that the wave numbers k and K are different in free space and the dielectric. (a) Write the electric and magnetic fields in vector form. (b) Write down the equations relating the fields to one another at the boundaries. (c) Write down the 4 linear equations for determining ER, E2, E3 and ET in terms of the "given" EI. (d) Bonus: Note that (for the fixed value of n=3) your result will depend only on the quantity Ka. Use a computer code to obtain the reflection coefficient |ER/EI|2 and plot it for 0 < Ka < 2.
Remember the transmission line constructed of two parallel plates? Find the capacitance and inductance per unit length, and use those quantities to determine the speed of potentials and currents associated with TEM waves as well as the characteristic impedance.
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
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