Chapter 21, Magnetic Fields Iv Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

(21.4) A permanent-magnet loudspeaker
Figure $21-7$ shows a cross section of a common form of magnetic circuit for a loudspeaker. The permanent magnet is a cylinder marked NS. The yoke has the form of a cup with a top plate, and the voice coil lies in the air gap with its axis vertical.
The top plate has a diameter of $25.0$ millimeters, and the gap is $2.50$ millimeters wide. The magnet is made of Alnico $V$, it is
$25.0$ millimcters long, its diameter is $20.0$ millimeters, and it operates at its. optimum $H$ of $4 \times 10^{4}$ ampere-turns/meter.

Calculate $B$ in the gap. The reluctance of the yoke is negligible. The magnetomotance of the permanent magnet is $4 \times 10^{4}$ times its length. The value of $B$ in loudspeakers is usually in the range of $0.3$ to 2 teslas.

Dading Chen

Numerade Educator

01:52

Problem 2

Magnetic circuit
A magnetic circuit comprises an air gap, as in Fig. $21-8$, with $R_{2}-R_{1} \ll$
$a$. Find an approximate expression for the reluctance of the gap.

Dominador Tan

Numerade Educator

02:45

Problem 3

Iron ring with a thin air gap
An iron ring carries a 300 -turn coil. The ring has a mean diameter of 400 millimeters, a cross section of 1000 millimeters $^{2}$. and a relative permeability of 500 .
(a) Calculate $B$ when the current in the coil is 1 ampere.

Aja S

Numerade Educator

01:01

Problem 4

(21.4) Plotting a magnetic field with an electrolytic tank.
Figure $21-9$ shows one example of the use of an electrolytic tank for plotting a magnetic field in a region where there are no currents and no magnetic materials. Here the electrodes are shaped like the pole pieces of an electromagnet. The model is strictly valid if the relative permittivity of the pole pieces is infinite.
The lines of $\boldsymbol{E}$ for the model are identical to the lines of $\boldsymbol{B}$ for the electromagnet. Let us see why.
(a) Which three differential equations does $B$ satisfy in free space? The equation $\boldsymbol{B}=\boldsymbol{\nabla} \times \boldsymbol{A}$ is not useful here. Which three differential equations does $\boldsymbol{E}$ satisfy in the model?
(b) Show that $\boldsymbol{B}$ is derivable from a potential: $\boldsymbol{B}=-\boldsymbol{\nabla} u$. The function $u$ is the scalar magnetic potential.
(c) Show that a surface of constant $u$ is orthogonal to the lines of $\boldsymbol{B}$.
Thus a surface of constant 4 corresponds to an equipotential.
(d) Show that $\boldsymbol{\nabla}^{2} u=0$.
(e) On the model, $\int_{c} \boldsymbol{E} \cdot \boldsymbol{d} l=V_{\prime}$ where $C$ is any curve that goes from one electrode to the other and $V$ is the applied voltage. What is the corresponding equation for $\boldsymbol{B}$ ?
(f) On the model, $I=\int \sigma \boldsymbol{E} \cdot d s 4$, where $I$ is the current between the electrodes, $\sigma$ the conductivity of the electrolyte, and $d \mathscr{A}$ an element of area on an electrode. What is the corresponding equation for the magnetic flux?
One could also deduce the reluctance from the resistance $V / I$.

Dominador Tan

Numerade Educator