Chapter 7: Q7.53P (page 349)

The current in a long solenoid is increasing linearly with time, so the flux is proportional $t$ :. $ϕ = α t$ 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 - $- \int_{a}^{b} E \times d l$ between the terminals and through the meter. [Answer: $V_{1} = α R_{1} / (R_{1} + R_{2})$ . Notice that $V_{1} \neq V_{2}$ , even though they are connected to the same points]

Short Answer

Expert verified

The expression for voltage across the resistance is $\frac{α R_{1}}{R_{1} + R_{2}}$ and is $- \frac{α R_{2}}{R_{1} + R_{2}}$ .

Step by step solution

Write the given data from the question.

The relationship between current and flux, $ϕ = α t$

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

Determine the formulas to calculate the voltmeter reading.

The expression for the induced emf is given as follows.

$ε = - \frac{d ϕ}{d t}$ ……. (1)

Draw the expression for the voltmeter reading.

Calculate the induced emf.

Substitute $α t$ for $ϕ$ into equation (1).

$\begin{array}{l} ε = \frac{d}{d t} (α t) \\ ε = α \end{array}$

The current in the registers is given by

$I = \frac{ε}{R_{1} + R_{2}}$

Substitute $α$ for $ε$ into above equation.

$I = \frac{α}{R_{1} + R_{2}}$

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

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

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

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

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

$V_{2} = - I R_{2}$

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

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

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Write the given data from the question.

Determine the formulas to calculate the voltmeter reading.

Draw the expression for the voltmeter reading.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Fields in Physics

Measurements

Quantum Physics

Fluids

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.