Chapter 14, Problem 3SP

A rectangular coil of wire has dimensions of 4 cm by 8 cm and is wound with 70 turns of wire. It is turned between the pole faces of a horseshoe magnet that produces an approximately uniform field of 0.5 T, so that sometimes the plane of the coil is perpendicular to the field and sometimes it is parallel to the field.

a.    What is the area bounded by the rectangular coil?

b.    What is the maximum value of the total magnetic flux that passes through the coil as it is turned?

c.    What is the minimum value of the total flux through the coil as it turns?

d.    If the coil makes one complete turn each second and turns at a uniform rate, what is the time involved in changing the flux from the maximum value to the minimum value?

e.    What is the average value of the voltage generated in the coil as it passes from the maximum to the minimum value of flux?

To determine

The area bounded by the rectangular coil.

Answer to Problem 3SP

The area of coil is $\text{32}{\text{cm}}^{\text{2}}$.

Explanation of Solution

Given Info Dimensions of rectangular coil is 4 cm by 8 cm, number of turns in coil is 70 and the magnetic field is $0.5\text{T}$.

Write the formula to calculate the area of coil.

$A=lb$

Here,

$A$ is the area of coil

$l$ is the length of coil

$b$ is the breadth of coil

Substitute 8 cm for $l$ and 4 cm for $b$ in the above equation to find $A$.

$\begin{array}{c}A=\left(\text{8 cm}\right)\left(\text{4 cm}\right)\\ =\text{32}{\text{cm}}^{\text{2}}\end{array}$

Conclusion:

Therefore, the area of coil is $\text{32}{\text{cm}}^{\text{2}}$.

To determine

The maximum value of magnetic flux through coil.

Answer to Problem 3SP

The maximum value of magnetic flux is $0.112\text{T}\cdot {\text{m}}^{\text{2}}$.

Explanation of Solution

Given Info Dimensions of rectangular coil is 4 cm by 8 cm, number of turns in coil is 70 and the magnetic field is $0.5\text{T}$.

Magnetic flux through the coil is maximum when the plane of coil is perpendicular to the magnetic field lines.

Write the formula to calculate the magnetic flux.

$\varphi =NBA\cos \theta$

Here,

$\varphi$ is the magnetic flux

$N$ is the number of turns

$B$ is the magnetic field

$\theta$ is the angle between plane of coil and magnetic field lines

Substitute 70 for $N$,$0.5\text{T}$ for $B$,$\text{32}{\text{cm}}^{\text{2}}$ for $A$ and $0°$ for $\theta$ in the above equation to find ${\varphi }_{\max }$.

$\begin{array}{c}{\varphi }_{\max }=\left(70\right)\left(0.5\text{T}\right)\left(\text{32}{\text{cm}}^{\text{2}}\left(\frac{\text{1}{\text{m}}^{\text{2}}}{{\text{10}}^{\text{4}}{\text{cm}}^{\text{2}}}\right)\right)\left(\cos 0°\right)\\ =0.112\text{T}\cdot {\text{m}}^{\text{2}}\end{array}$

Conclusion:

Therefore, the maximum value of magnetic flux is $0.112\text{T}\cdot {\text{m}}^{\text{2}}$.

To determine

The minimum value of magnetic flux through coil.

Answer to Problem 3SP

The minimum value of magnetic flux is $0.112\text{T}\cdot {\text{m}}^{\text{2}}$.

Explanation of Solution

Given Info Dimensions of rectangular coil is 4 cm by 8 cm, number of turns in coil is 70 and the magnetic field is $0.5\text{T}$.

Magnetic flux through the coil is minimum when the plane of coil is parallel to magnetic field lines.

Write the formula to calculate the magnetic flux.

$\varphi =NBA\cos \theta$

Substitute 70 for $N$,$0.5\text{T}$ for $B$,$\text{32}{\text{cm}}^{\text{2}}$ for $A$ and $90°$ for $\theta$ in the above equation to find ${\varphi }_{\min }$.

$\begin{array}{c}{\varphi }_{\min }=\left(70\right)\left(0.5\text{T}\right)\left(\text{32}{\text{cm}}^{\text{2}}\left(\frac{\text{1}{\text{m}}^{\text{2}}}{{\text{10}}^{\text{4}}{\text{cm}}^{\text{2}}}\right)\right)\left(\cos 90°\right)\\ =0\text{T}\cdot {\text{m}}^{\text{2}}\end{array}$

Conclusion:

Therefore, the minimum value of magnetic flux is $0\text{T}\cdot {\text{m}}^{\text{2}}$.

To determine

The time required for change in flux.

Answer to Problem 3SP

The time required for flux change is $0.25\text{s}$.

Explanation of Solution

Given Info: The current through the wires are $8\text{A}$ and $12\text{A}$, separation between wires is $4\text{cm}$ and coil makes one complete turn in one second.

Field lines become perpendicular and parallel to plane of loop two times during one complete turn of coil. This means that during one complete turn, change in maximum value of flux to minimum occurs four times.

Write the formula to calculate the time required for flux change.

$t=\frac{T}{n}$

Here,

$t$ is the time taken for flux change

$T$ is the time taken by coil for one complete turn

$n$ is the number of times which flux changes from maximum to minimum in one complete turn

Substitute $1\text{s}$ for $T$ and $4$ for $n$ in the above equation to find $t$.

$\begin{array}{c}t=\frac{1\text{s}}{4}\\ =0.25\text{s}\end{array}$

Conclusion:

Therefore, time taken for flux change is $0.25\text{s}$.

To determine

The average value of induced voltage inn coil.

Answer to Problem 3SP

The average value of induced voltage is $0.448\text{V}$.

Explanation of Solution

Write the Faraday’s law to calculate the average value of induced voltage.

$\epsilon =\frac{{\varphi }_{\max }-{\varphi }_{\min }}{t}$

Here,

$\epsilon$ is the average value of induced voltage

Substitute $0.112\text{T}\cdot {\text{m}}^{\text{2}}$ for ${\varphi }_{\max }$,$0\text{T}\cdot {\text{m}}^{\text{2}}$ for ${\varphi }_{\min }$ and $0.25\text{s}$ for $t$ in the above equation to find $\epsilon$.

$\begin{array}{c}\epsilon =\frac{0.112\text{T}\cdot {\text{m}}^{\text{2}}-0\text{T}\cdot {\text{m}}^{\text{2}}}{0.25\text{s}}\\ =0.448\text{V}\end{array}$

Conclusion:

Therefore, the average value of induced voltage is $0.448\text{V}$.