(a) Explanation Write the expression for Maxwell’s fourth equation for a capacitor. ∮ B → ⋅ d l → = μ 0 ε 0 d ϕ E d t (I) Here, ∮ B → ⋅ d l → is the line integral of magnetic field over a closed loop, μ 0 is the permeability of free space, ε 0 is the permittivity of free space, d ϕ E d t is the rate of change of electric flux and t is the time. Substitute B ( 2 π R path ) for ∮ B → ⋅ d l → and π R f l u x 2 E for ϕ E in equation (I). B ( 2 π R path ) = μ 0 ε 0 d d t ( π R f l u x 2 E ) Here, R path is the radius of Amperian loop, R flux is the radius through which the electric field flux is bounded and E is the electric field. Subscripts are used on the radial variable because there might not be electric field flux through the entire area bounded by the Amperian path. Rearrange the above expression for B . B = μ 0 ε 0 π R f l u x 2 2 π R path d E d t (II) Here, B is the magnetic field, μ 0 is the permeability of free space and ε 0 is the permittivity of free space. Write the expression for electric flux as. E = V d Here, E is the electric field, V is the voltage and d is the distance between plates. Differentiate the above expression with respect to t . d E d t = 1 d d V d t (III) Substitute V 0 sin ( 2 π f t ) for V in the equation (III). d E d t = 1 d d d t [ V 0 sin ( 2 π f t ) ] Simplify the above expression. d E d t = 2 π f V 0 d cos ( 2 π f t ) (IV) Here, V 0 is the peak value of voltage and f is the frequency (b) To determine Expression for the amplitude of B ¯ of time dependent field when R ≤ R 0 and when R ≥ R 0 . (c) To determine Plot B ¯ for the range 0 ≤ R ≤ 10 cm .

Physics for Scientists and Engineers with Modern Physics

4th Edition

ISBN: 9780131495081

Author: Douglas C. Giancoli

Publisher: Addison-Wesley

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Question

Chapter 31, Problem 7P

(a)

To determine

Show that in the region between the plates the magnitude of the induced magnetic field is given by B=B0(R)cos(2πft).

(b)

To determine

Expression for the amplitude of B¯ of time dependent field when R≤R0 and when R≥R0.

(c)

To determine

Plot B¯ for the range 0≤R≤10 cm.

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Chapter 31, Problem 6P

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Chapter 31 Solutions

Physics for Scientists and Engineers with Modern Physics

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Knowledge Booster

Show that in the region between the plates the magnitude of the induced magnetic field is given by B = B 0 ( R ) cos ( 2 π f t ) .

Solution Summary: The author demonstrates the magnitude of the induced magnetic field in the region between the plates. Write the expression for Maxwell’s fourth equation for a capacitor.

Show that in the region between the plates the magnitude of the induced magnetic field is given by B=B0(R)cos(2πft).

Expression for the amplitude of B¯ of time dependent field when R≤R0 and when R≥R0.

Plot B¯ for the range 0≤R≤10 cm.

Want to see the full answer?

Chapter 31 Solutions