## PHYS208 Fundamentals of Physics II

### 1.

 Typical ac power (rms emf = 120 V, frequency = 60 Hz) is being used to drive a single-phase motor (L = 40 mH, R = 10 ohm) as shown in Fig. 1. Transmission losses are modeled with an additional resistance (r = 5.0 ohm). Evaluate the following quantities: a) rms current: b) power supplied by emf to R: c) What value of capacitance should be added, as shown in Fig. 2, to maximize the power transferred to the motor? Re-evaluate the following quantities: d) rms current: e) power supplied by emf to R: g. Sketch (to scale) a phasor diagram depicting the emf and the voltages across the inductor and capacitor.

### 2.

Although the following problem concerns electromagnetic waves and transmission lines, detailed knowledge of those topics is not required here. It has been included to test your ability to manipulate physical quantities and investigate new relationships.

Coaxial cable, considered in the two previous exams, is used to transfer high frequency signals between electronic test instruments. RG58/U cable consists of an inner wire of diameter a = 0.81 mm, surrounded by polyethlene (a dielectric,  = 2.25), and a concentric conducting braid of diameter b = 2.9 mm. The capacitance per unit length C0 is 2    / ln(b/a); the inductance per unit length L0 is (/2) ln(b/a).

a) Express the quantity (L0 C0-1/2 as a fraction of the speed of light in vacuum c:

b) Does the speed in part a depend on the geometric parameters, the dielectric constant, or both?

c) Evaluate numerically the quantity ( / )1/2. (Express in ohms!) This quantity is known as the characteristic impedance of free space.

d) Express the quantity (L0 / C0)1/2 as a fraction of the free space impedance.

e) Does the characteristic impedance in part d depend on the geometric parameters, the dielectric constant, or both?

f) Evaluate numerically the characteristic impedance for RG58/U coax cable.

### 3.

The transmission line geometry being considered this semester [Fall 1993] in PHYS208 is the so-called twin-lead, often seen in use as a lead-in wire from a TV antenna. It consists of 2 long parallel wires of radius a with centers separated by distance d, held in place by a material with dielectric constant .

a) Derive an expression for the capacitance per unit length C0. Your solution will be judged, in part, on its thoroughness and clarity.

b) For a = 0.50 mm, d = 8.0 mm, and  = 2.25, evaluate C0.

### 4.

 Consider the circuit shown. a) What is the emf of battery 1? b) What is the terminal voltage of battery 2? c) What is the power dissipated by the 12 ohm resistor? d) What is the power supplied by the emf of battery 1?

### 5.

 Consider the circuit shown in which the capacitor is initially uncharged. Now imagine the switch to be closed. a) Determine the initial current. b) Determine the charge Qmax on the capacitor when it is fully charged. c) Find the time required for the charge on the capacitor to reach 99% of Qmax. d) Find the time required for the energy stored in the capacitor to reach 99% of its final value.

### 6.

 The circular plates depicted to the left have a radius R. Use Ampere's law as modified by Maxwell to derive an expression for the magnetic field at a point between the plates at a distance r from the axis of the plates when the current into the positive plate is i. You may assume a uniform field between plates; that is, assume a parallel plate capacitor with large plates.

### 7.

 Your solution will be judged, in part, on its thoroughness and clarity. a. Apply Ampere's law to derive an expression for the magnetic field inside a rectangular toroid (half shown). b. Derive an expression for the inductance of a toroid, using the result of part a.

### 8.

 Consider an RLC series circuit designed for 60 Hz, 120 Vrms operation. On a trip to Europe (f = 50 Hz) the appliance fails. Why? Hint: Analyze circuit completely for both frequencies.

### 9.

a.  Specify the condition for electromagnetic resonance in an RLC series circuit in five "different" ways.

b.  Under what condition will the terminal voltage of a battery be greater than its emf?

c.  When a dielectric is inserted into a capacitor that is held at a fixed potential difference by a battery, does the charge stored increase, decrease, or remain the same? Briefly defend your answer.

d.  A proton moves horizontally to the West through a magnetic field that is downward. Is the force on the proton directed to the North, South, East, West, upward or downward?

e.  For a series RLC circuit which of the following is not a condition of resonance:

1. XL = XC;
2. tangent of the phase angle is unity;
3. angular frequency of the emf is (LC)-1/2;
4. impedance is purely resistive.

f.  To match the impedance of a load with the source, we want to:

1. minimize total reactance;
2. minimize total impedance;