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Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

Chapter 7: Q7.53P (page 349)

The current in a long solenoid is increasing linearly with time, so the flux is proportional $t$ :. $蠒 = 伪 t$ Two voltmeters are connected to diametrically opposite points (A and B), together with resistors ( $R_{1}$ and $R_{2}$ ), as shown in Fig. 7.55. What is the reading on each voltmeter? Assume that these are ideal voltmeters that draw negligible current (they have huge internal resistance), and that a voltmeter register - $- {鈭�}_{a}^{b} E 脳 d l$ between the terminals and through the meter. [Answer: $V_{1} = 伪 R_{1} / (R_{1} + R_{2})$ . Notice that $V_{1} 鈮� V_{2}$ , even though they are connected to the same points]

Short Answer

Expert verified

The expression for voltage across the resistance is $\frac{伪 R_{1}}{R_{1} + R_{2}}$ and is $鈭� \frac{伪 R_{2}}{R_{1} + R_{2}}$ .

Step by step solution

01

Write the given data from the question.

The relationship between current and flux, $蠒 = 伪 t$

The two registers are $R_{1}$ and $R_{2}$ .

02

Determine the formulas to calculate the voltmeter reading.

The expression for the induced emf is given as follows.

$蔚 = - \frac{d 蠒}{d t}$ 鈥︹€�. (1)

03

Draw the expression for the voltmeter reading.

Calculate the induced emf.

Substitute $伪 t$ for $蠒$ into equation (1).

$\begin{array}{l} 蔚 = \frac{d}{d t} (伪 t) \\ 蔚 = 伪 \end{array}$

The current in the registers is given by

$I = \frac{蔚}{R_{1} + R_{2}}$

Substitute $伪$ for $蔚$ into above equation.

$I = \frac{伪}{R_{1} + R_{2}}$

The voltage across the register $R_{1}$ is given by,

$V_{1} = I R_{1}$

Substitute $\frac{伪}{R_{1} + R_{2}}$ for $I$ into above equation.

$\begin{array}{l} V_{1} = \frac{伪}{R_{1} + R_{2}} R_{1} \\ V_{1} = \frac{伪 R_{1}}{R_{1} + R_{2}} \end{array}$

The voltage across the register $R_{1}$ is given by,

$V_{2} = 鈭� I R_{2}$

Substitute $\frac{伪}{R_{1} + R_{2}}$ for $I$ into above equation.

$\begin{array}{l} V_{2} = 鈭� \frac{伪}{R_{1} + R_{2}} R_{2} \\ V_{2} = 鈭� \frac{伪 R_{2}}{R_{1} + R_{2}} \end{array}$

Hence, the expression for voltage across the resistance $R_{1}$ is $\frac{伪 R_{1}}{R_{1} + R_{2}}$ and $R_{2}$ is $鈭� \frac{伪 R_{2}}{R_{1} + R_{2}}$ .

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Most popular questions from this chapter

Two concentric metal spherical shells, of radius a and b, respectively, are separated by weakly conducting material of conductivity $蟽$ (Fig. 7 .4a).

(a) If they are maintained at a potential difference V, what current flows from one to the other?

(b) What is the resistance between the shells?

(c) Notice that if b>>a the outer radius (b) is irrelevant. How do you account for that? Exploit this observation to determine the current flowing between two metal spheres, each of radius a, immersed deep in the sea and held quite far apart (Fig. 7 .4b ), if the potential difference between them is V. (This arrangement can be used to measure the conductivity of sea water.)

Find the self-inductance per unit length of a long solenoid, of radius R , carrying n turns per unit length.

Refer to Prob. 7.16, to which the correct answer was

$E (s, t) = \frac{渭_{0} I_{0} 蝇}{2 蟿蟿} s i n (蝇 t) I n (\frac{a}{s}) \hat{z}$

(a) Find the displacement current density $J_{d}$ 路

(b) Integrate it to get the total displacement current,

$I_{d} = 鈭� J_{d} . d a$

Compare $I_{d}$ and I. (What's their ratio?) If the outer cylinder were, say, 2 mm in diameter, how high would the frequency have to be, for $I_{d}$ to be 1% of I ? [This problem is designed to indicate why Faraday never discovered displacement currents, and why it is ordinarily safe to ignore them unless the frequency is extremely high.]

A toroidal coil has a rectangular cross section, with inner radius a , outer radius $a + w$ , and height h . It carries a total of N tightly wound turns, and the current is increasing at a constant rate $(dl / dt = k)$ . If w and h are both much less than a , find the electric field at a point z above the center of the toroid. [Hint: Exploit the analogy between Faraday fields and magnetostatic fields, and refer to Ex. 5.6.]

A circular wire loop (radius r , resistance R ) encloses a region of uniform magnetic field, B , perpendicular to its plane. The field (occupying the shaded region in Fig. 7.56) increases linearly with time $(B = 鈭� t)$ An ideal voltmeter (infinite internal resistance) is connected between points P and Q.

(a) What is the current in the loop?

(b) What does the voltmeter read? Answer: $[鈭� r^{2} / 2]$

See all solutions

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