Chapter 14, Problem 3SP

(a)

To determine

The area bounded by the rectangular coil.

Answer to Problem 3SP

The area bounded by the rectangular coil is $0.0018{\text{m}}^2$.

Explanation of Solution

Given info: The length of rectangular coil is $6\text{cm}$ and breadth of coil is $3\text{cm}$.

The coil provided is rectangular in shape. Therefore area of coil is found by multiplying the breadth and length.

Write the equation to find the area of a rectangular loop.

$A=lb$

Here,

$A$ is the area of the rectangular coil

$l$ is the length of the rectangular coil

$b$ is the breadth of the rectangular coil

Substitute $6\text{cm}$ for $l$ and $3\text{cm}$ for $b$ in the equation to get $A$.

$\begin{array}{c}A=6\text{cm}\left(\frac{1\text{m}}{100\text{cm}}\right)\times 3\text{cm}\left(\frac{1\text{m}}{100\text{cm}}\right)\\ =0.0018{\text{m}}^2\end{array}$

Conclusion:

Therefore, the area of the rectangular coil is $0.0018{\text{m}}^2$.

(b)

To determine

The maximum value of magnetic flux passing through the rectangular coil as it is turned.

Answer to Problem 3SP

The maximum value of magnetic flux that is passing through the rectangular coil is $0.043{\text{Tm}}^2$.

Explanation of Solution

Given info: Magnetic field applied is $0.4\text{T}$ , area of rectangular coil is $0.0018{\text{m}}^2$ , number of turns in the rectangular coil is $60$.

Write the equation to find the magnetic flux passing through an area.

$\varphi =NBA\cos \theta$

Here,

$\varphi$ is the magnetic flux passing through the rectangular coil

$N$ is the number of turns in the rectangular coil

$B$ is the magnetic field intensity applied

$A$ is the area of the rectangular coil

$\theta$ is the angle between the rectangular coil and the magnetic field

Maximum value of magnetic flux passes through the coil when the magnetic field and the rectangular coil are parallel to each other. Therefore the angle between rectangular coil and magnetic field $\theta$ is $0°$.

Substitute $60$ for $N$ , $0.4\text{T}$ for $B$ , $0.0018{\text{m}}^2$ for $A$ ,and $0°$ for $\theta$ in the equation to get $\varphi$.

$\begin{array}{c}\varphi =60\times 0.4\text{T}\times 0.0018{\text{m}}^2\cos 0°\\ =0.0432{\text{Tm}}^2\end{array}$

Conclusion:

Therefore, the maximum value of magnetic flux that passes through the rectangular coil is $0.043{\text{Tm}}^2$.

(c)

To determine

The minimum value of flux that passes through the rectangular coil as it is rotated.

Answer to Problem 3SP

The minimum value of magnetic flux that passes through the rectangular coil is zero.

Explanation of Solution

Given info: Magnetic field applied is $0.4\text{T}$ , area of rectangular coil is $0.0018{\text{m}}^2$ , number of turns in the rectangular coil is $60$.

Write the equation to find the magnetic flux passing through an area.

$\varphi =NBA\cos \theta$

Here,

$\varphi$ is the magnetic flux passing through the rectangular coil

$N$ is the number of turns in the rectangular coil

$B$ is the magnetic field intensity applied

$A$ is the area of the rectangular coil

$\theta$ is the angle between the rectangular coil and the magnetic field

Minimum value of magnetic flux passes through the coil when the magnetic field and the rectangular coil are perpendicular to each other. Therefore the angle between rectangular coil and magnetic field $\theta$ is $90°$.

Substitute $60$ for $N$ , $0.4\text{T}$ for $B$ , $0.0018{\text{m}}^2$ for $A$ ,and $90°$ for $\theta$ in the equation to get $\varphi$.

$\begin{array}{c}\varphi =60\times 0.4\text{T}\times 0.0018{\text{m}}^2\cos 90°\\ =0{\text{Tm}}^2\end{array}$

Conclusion:

Therefore, the minimum value of magnetic flux that passes through the rectangular coil is zero.

(d)

To determine

The time taken to change the flux from a maximum to minimum in a complete cycle.

Answer to Problem 3SP

The time taken for a change from maximum flux to minimum flux is $0.25\text{s}$.

Explanation of Solution

A complete turn of the rectangular coil takes one second. A complete turn means a $360°$ turn of the rectangular coil. During a complete turn the coil experiences three maximum and two minimums.

The first change from maximum to minimum flux occurs between the $0°$ to $90°$ change. Therefore a change from maximum to minimum flux occurs during every $90°$ rotation. In a complete cycle there are four $90°$ changes. Therefore the time required for one of such change is found by dividing the time taken by a complete rotation into four rotations.

Time for one complete rotation is $1\text{sec}$. Therefore time for a change from a maximum flux to minimum flux is $0.25\sec$

Conclusion:

Therefore, the time taken for a change from maximum flux to a minimum flux is $0.25\text{sec}$.

(e)

To determine

Maximum value of voltage developed in the coil when the flux changes from a maximum to a minimum.

Answer to Problem 3SP

The maximum value of voltage developed when the flux changes from maximum value to minimum value is $0.1728\text{V}$.

Explanation of Solution

Write the equation to find the voltage developed when magnetic flux changes from maximum value to minimum value.

$\in =\frac{d\varphi }{dt}$

Here,

$\in$ is the voltage developed in the rectangular coil

$d\varphi$ is the change in flux from a maximum to a minimum

$dt$ is the time taken for the change in value of flux

$d\varphi$ is the difference between the maximum flux value and minimum flux value.

$\begin{array}{c}d\varphi =0.043{\text{Tm}}^2-0{\text{Tm}}^2\\ =0.043{\text{Tm}}^2\end{array}$

$dt$ is the time taken for a change between a maximum flux to a minimum flux value which is equal to $0.25\text{s}$.

Substitute $0.043{\text{Tm}}^2$ for $d\varphi$ , $0.25\text{s}$ for $dt$ in the equation to find $\in$.

$\begin{array}{c}\in =\frac{0.043{\text{Tm}}^2}{0.25\text{s}}\\ =0.1728\text{V}\end{array}$

Conclusion:

Therefore, the maximum value of voltage induced when flux changes from a maximum value to a minimum value is $0.1728\text{V}$.