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University of Santo TomasCollege of Science, Department of Mathematics and Physics

Physics 318, Assignment 2

Instructions: Write down complete calculations with correct units.

1. Two loops with a mutual inductance of 85 H are arranged close to each other. The currentin loop one changes with a constant rate of 35 mA/s. What emf is induced in loop two?

2. The current in a loop with a self inductance of 90 H shall be changed with a constant rateof 50 mA/s. What back emf is induced in this loop?

3. Applying Kirchhoffs loop rule to a R-L circuit with resistor (resistance R), inductor(inductanceL), and battery (emfE0) in series, one gets the differential equation E0 IR

LdI/dt= 0 for the current I.(A) Show that I(t) = (1 exp(Rt/L))E0/R is the solution of the differential equation forthe condition, that the initially open circuit is closed at t= 0.(B) Draw the current versus time (t 0) graph for this solution.(C) Show that I(t) = I0exp(Rt/L) is the solution of the differential equation for thecondition, that the initially closed circuit with current I0 is opened at t = 0.(D) Draw the current versus time (t 0) graph for this solution.

4. A R-L circuit with a resistor (resistance R = 30.0 ) and an inductor (inductanceL =120.0 H) is connected in series to a battery (E0 = 12 V). The initially open circuit is closedat timet= 0.

(A) What is the value of the current at time t= 0.5 s.(B) When does the current in the circuit reach the value E0/(3R)?(C) When does the current in the circuit reach the value E0/R?

5. Applying Kirchhoffs loop rule to a L-Ccircuit with an inductor (inductance L) and acapacitor (capacitance C) in series, one gets the equation LdI/dt Q/C = 0. With thedefinition of the currentI=dQ/dt, this becomes a differential equation d2Q/dt2+Q/(LC) =0 for the charge Q on the capacitor.(A) Show that the sinusoid Q(t) =Q0cos(t+) is the solution of the differential equationfor the condition, that the maximum charge on the capacitor is Q0 ( is a phase constant,

and the angular frequency of the oscillating charge on the capacitor).(B) From part (A) follows a relationship between the inductance L as well as the capacitanceCand the angular frequency . What is this relationship?(C) Derive the formula for the corresponding alternating current I(t).(D) Draw a current versus time graph for this alternating current.

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6. The continuity equation for the charges can be derived from two out of the four Maxwellequations.(A) Write down these two equations.(B) Derive the continuity equation from them.(C) What does the continuity equation tell us about a changing total charge within a volume

and the corresponding net current flowing through the surface of this volume?

7. How is the Poynting vector S defined, what physical quantity does it describe, and inwhat conservation laws does it play a role?

8.. Is the function f(z, t) = A exp[k(z vt)] a solution of the wave equation 2f/z2 =v22f/t2 (mathematical proof required)?

9. A sinusoidal wave has a wavelength and a velocity v. How can one calculate the wavenumber k, frequency f, angular frequency , and periodT from and v?

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