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1

Tokyo Institute of Technology Graduate School of Science and Engineering

Department of Electrical and Electronic Engineering

Department of Physical Electronics

Entrance Examination held on 2010/8/17 for 2010 Oct./2011 Apr.

Electromagnetism

Dept. of Electrical and Electronic EngineeringDept. of Physical Electronics

13:30 – 15:00

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Subject

Electromagnetism

1． A circle and a sector form are joined together in a fan-like shape as shown in Fig. 1.1. The

radius of the circle is a and the radius of the sector form is 5a with a center angle of / 2 . Fig 1.2

shows a parallel-plate capacitor which consists of plate A and B , each having the same shape as

shown in Fig. 1.1. The distance between the plates is ( )d d a= and the plate B can rotate on the

central axis O of the circle. The capacitance ( ) C of the parallel-plate depends only on the hatching

area in Fig.2 by neglecting fringing effect at the edges of the plates. The parallel-plate is in vacuum

and the electric permittivity is0

. Answer the following questions.

1) Find the capacitance0

C at 0 , and the capacitance ( ) C for 02

.

2) Plate B rotates from the 0 to 2 . Sketch the capacitance ( ) C as a function of in

the answer sheet. Then determine the maximum and minimum values of ( ) C usingcapacitance

0C .

Assuming that a total charge Q on the plate A and Q on the plate B are uniformly distributed

over the plates at 0 . Answer the following questions: 3) ~ 5).

3) The plate B is forced to rotate a small angle of ( 0) . Explain the force applied to the plate

B in the direction by using the principle of virtual displacement.

4) The plate B rotates from the angle of 0 to 2 . Find the electrostatic energy stored in

the capacitor ( ) C at 0 and repeat at . If there is a difference between them, show

why such a difference is observed.

5) Under the conditions of problem 4), find the voltage ( )V between the plates during the rotation

of plate B . Then, sketch the outline of ( )V in the answer sheet.

Fig. 1.1 Fig. 1.2

2

5a

a

O

A

B

O

【Electromagnetism】

（Page 1 of 4 pages）

8/13/2019 H23-pmENG


Subject

Electromagnetism

The parallel-plate capacitor is installed in a circuit as shown in Fig. 1.3. The variable capacitor ( ) C

represents the parallel-plate capacitor. One terminal of ( ) C is connected to the voltage source

1V 1 0V and the other is grounded through the diode

1 D . pV is the voltage at the connection point

between ( ) C and1

D , and the point is connected to the voltage source2

V 2 0V through the

switch S and the diode2

D . The polarity of each source is as shown in Fig. 1.3. Assume that each

diode is zero impedance with forward voltage and infinite impedance with reversed voltage. In the

initial conditions, the switch is open, the angle of plate B is 0 , and the capacitor ( ) C is

charged up to a constant voltage through the diode1

D . Answer the following questions: 6) ~ 8).

6) The plate B rotates to the angle of . Find the minimum valuemin

V and the maximum

valuemax

V of the voltage pV during the rotation.

7) After closing the switch S , the plate B rotates form the angle of 0 to 2 . The voltage

maxV determined in problem 6) is as max 2V V . Sketch the stored charge ( ) Q in ( ) C versus

the voltage ( ) V of ( ) C . Then explain the derivation-processes.

8) Under the conditions of problem 7), find the transferred energy2

W (= transferred charge×

voltage) to the voltage source2

V and the transferred energy1

W from the capacitor ( ) C to the

voltage source1V during the rotation from the angle of 0 to 2 . Then calculate

2 1W W and explain what the subtraction means.

Fig. 1.2 (same figure as shown in page 1) Fig. 1.3

S

2V

( ) C p

1V

1 D

2 D

【Electromagnetism】

（Page 2 of 4 pages）

A

B

O

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