October 12, 1992 Midterm Exam 1
This is a closed book exam; neither notes nor formula cards (including those stored in programmable calculators) may be referred to during the exam.
Since the exam booklet is separated for grading; it is important to:
Please read questions carefully.
Credit may be lost inadvertently if solutions are not neat and orderly.
Be careful with units, signs, and significant figures.
Some possibly useful constants:
1. 30 points -- Short-answer questions:
a. Consider two isolated, uniform spheres (radius: 1 cm), one conducting and the other not, charged so that both are at 15 V.
b. Write the formula for Gauss's law and then state it in words, carefully and concisely.
c. An electron is released from rest in an electric field. Will it move in the direction of increasing or decreasing potential? Briefly justify your answer.
d. Two capacitors are identical except that the region between the plates of capacitor A is filled with a dielectric and B is empty. These capacitors are connected in parallel and maintained at the same potential difference by a battery. Is the electric field intensity in the region between the plates of capacitor A smaller, the same, or larger than that in B? Be careful with your reasoning.
2. 30 points
Coaxial cable is typically used to make high-speed circuit connections between electronic test instruments. RG58/U cable consists of an inner wire of diameter a = 0.81 mm, surrounded by polyethylene (a dielectric, kappa = 2.25), a concentric conducting braid of diameter b = 2.9 mm, and insulation, as shown in the figure. Evaluate the capacitance per meter length.
(Please note that you may suppress the presence of the dielectric in your analysis until part c, where kappa may be introduced as a multiplicative factor.)
a. Use Gauss's law to determine the electric field at a point between long coaxial conductors when oppositely charged.
b. Use the result of part a to find the corresponding potential difference between the conductors.
c. Express the capacitance per length L of a coaxial cable in terms of the cable parameters: a, b, and kappa.
d. Evaluate the capacitance per meter length of RG58/U cable.
Consider the following circuit that represents a typical flashlight, two D-cells in series with a lamp.
Each D-cell provides an emf of 1.5 V and has an internal resistance r of 0.2 ohms.
The PR-2 lamp resistance R is 7.0 ohms; each battery is rated at 4 A-hr, a measure of its usable energy content.
a. What is the current in the circuit?
b. What is the power dissipated by the lamp?
c. What is the terminal voltage of either D-cell?
d. Estimate how long the flashlight will burn brightly.
4. 20 points -- Same circuit element as 3.
An inquisitive child finds an additional D-cell and exuberantly sets out to increase the brightness of the flashlight by hooking the third battery directly across the lamp as shown. Much to the child's chagrin, the lamp shines less brightly than in Question 3.
Analyze the circuit using the specified currents.
a. Apply the junction theorem at point A.
b. Apply the loop theorem on the left loop, clockwise from point A.
c. Apply the loop theorem on the right loop, counterclockwise from point A.
d. Solve for the currents i1, i2, and the current through the lamp.
e. What is the power now dissipated by the lamp?
f. What is the terminal voltage of the third D-cell added?