Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

Suppose that a parallel-plate capacitor has circular plates with a radius $R = 30 mm$ and, a plate separation of $5.00 mm$ . Suppose also that a sinusoidal potential difference with a maximum value of $150 V$ and, a frequency of $60 Hz$ is applied across the plates; that is,
$V = (150 V) sin [2 π (60 Hz) t]$
(a) Find $B_{m a x} (R)$ , the maximum value of the induced magnetic field that occurs at $r = R$ .
(b) Plot $B_{\max} (r)$ for $0 < r < 10 cm$ .

The maximum value of the induced magnetic field that occurs at $r = R$ is $B = 1.9 \times 10^{- 12} .$
The plot is given in the calculation section.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given

The radius of plates, $R = 30 mm = 0.03 m$

Plate separation, $d = 0.005 m$

Maximum potential difference, $V = 150 V$

Frequency, $f = 60 Hz$

$V = (150) \sin [2 π (60 Hz) t]$

Step 2: Determining the concept

By using the Maxwell equation, finding the magnetic field for the maximum potential, and plotting the graph maximum B vs r Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism.

Maxwell’s law of Induction-

$. \vec{B} . d \vec{A} = μ_{0} E_{0} A \frac{d E}{d t}$

The electric field is given as-

$E = \frac{V}{d}$

Where, B is the magnetic field, is the area enclosed by the Amperian loop, E is the electric field, t is the time, V is the potential difference and, d is the distance.

Step 3: (a) Determining the maximum value of the induced magnetic field that occurs at r=R

The electric field is given as-

$E = \frac{V}{d}$

The magnetic field induced by the changing electric field is given by the relation,

localid="1663162361810" style="max-width: none; vertical-align: -15px;" $\vec{B} . d \vec{A} = μ_{0} E_{0} A \frac{d E}{d t}$

Where, localid="1663162418635" $A$ is the area enclosed by the Amperian loop, which is, localid="1663162390281" $A = π d^{2}$ .

So that, for $r < R$ ,

$\begin{array}{rcl} B (2 π r) & = & μ_{0} E_{0} π r^{2} \frac{d E}{d t} \\ B & = & \frac{μ_{0} E_{0} r}{2} \frac{d E}{d t} \end{array}$

But,

$E = \frac{V}{d}$

So,

$\begin{array}{rcl} B & = & \frac{μ_{0} E_{0} r}{2 d} \frac{d V}{d t} \\ B & = & \frac{μ_{0} E_{0} r}{2 d} \frac{d}{d t} (V_{m a x} \sin ω t) \\ B & = & \frac{μ_{0} E_{0} r}{2 d} V_{m a x} ω \cos ω t \end{array}$

For the maximum value of potential $V_{m a x} = 150 V$ ,

$B = \frac{μ_{0} E_{0} r}{2 d} V_{m a x} ω$

The $r = R = 0.03 m$

localid="1663161555184" $\begin{array}{rcl} B & = & \frac{(4 π \times 10^{- 7} H/m) (8.85 \times 10^{- 12} F/m) \times 0.03 m}{2 \times 0.005 m} \times 150 \times 2 π \times 60 Hz \\ B & = & 1.9 \times 10^{- 12} T \end{array}$

Therefore, the maximum value of the induced magnetic field that occurs at $r = R$ is $B = 1.9 \times 10^{- 12} T .$

Step 4: (b) Determining the required plot

The maximum value of $B$ , the magnetic field is induced by the changing electric field so that,

$\begin{array}{rcl} B_{m a x} & = & {(\frac{μ_{0} E_{0} R^{2}}{2 r} \frac{d E}{d t})}_{m a x} \\ B_{m a x} & = & {(\frac{μ_{0} E_{0} R^{2}}{2 r d} \frac{d V}{d t})}_{m a x} \\ B_{m a x} & = & {(\frac{μ_{0} E_{0} R^{2}}{2 r d} V_{m a x} ω \cos ω t)}_{m a x} \\ B_{m a x} & = & {(\frac{μ_{0} E_{0} R^{2}}{2 r d} V_{m a x} ω)}_{m a x} \end{array}$

So, all values are constant except r.

B is dependent on the value of r, so plot the graph B vs r.

$\begin{array}{rcl} B_{m a x} & = & \frac{(4 π \times 10^{- 7} H/m) (8.85 \times 10^{- 12} F/m) \times {(0.03 m)}^{2}}{2 \times 0.005 m×r} \times 150 \times 2 π \times 60 Hz \\ = & 5.7 \times 10^{- 14} \times \frac{1}{r} \end{array}$

Here, r varies from 0 to 0.1.

Hence, plotted the graph $B_{m a x}$ vs $r$ , for varying from 0 to 0.1.

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Given

Step 2: Determining the concept

Step 3: (a) Determining the maximum value of the induced magnetic field that occurs at r=R

Step 4: (b) Determining the required plot

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