Chapter 20, Problem 38E

(a)

To determine

The inductance of the coil.

Answer to Problem 38E

The inductance of the coil is $394.78\text{H}$.

Explanation of Solution

Write the expression for the inductance in a coil.

    $L=4\pi k_m{k}^{\prime }{n}^{2}Al$        (1)

Here, $L$ is the inductance, $k_m$ is the magnetic constant, $k^{\prime }$ is a constant, $n$ is the number of turns per unit length, $A$ is the area of the coil and $l$ is the length.

Substitute $\left(\pi r^2\right)$ for $A$ and $\left(\frac{N}{l}\right)$ for $n$ in equation (1).

    $L=4\pi k_m{k}^{\prime }{\left(\frac{N}{l}\right)}^2\left(\pi r^2\right)l$        (2)

Here, $N$ is the number of turns of the coil and $r$ is the radius of the coil.

Conclusion:

Substitute $1$ for $k_m$, $\left({10}^{-7}\text{T}\text{m}{\text{A}}^{\text{-1}}\right)$ for $k^{\prime }$, $1000$ for $N$, $10\text{m}$ for $l$ and $100\text{m}$ for $r$ in equation (2).

    $\begin{array}{c}L=4\pi \left(1\right)\left(\left({10}^{-7}\text{T}\text{m}{\text{A}}^{\text{-1}}\right)\right){\left(\frac{1000}{100\text{m}}\right)}^2\left(\pi {\left(100\text{m}\right)}^2\right)\left(100\text{m}\right)\\ =394.78\text{H}\end{array}$

Thus, the inductance of the coil is $394.78\text{H}$.

(b)

To determine

The energy stored in the coil

Answer to Problem 38E

The energy is $4.4\times {10}^{12}\text{J}$.

Explanation of Solution

Write the expression for the energy stored in the coil.

    $U=\frac{1}{2}L{i}^2$        (3)

Here, $U$ is the energy, $L$ is the inductance and $i$ is the current.

Conclusion:

Substitute $394.78\text{H}$ for $L$ and $1.5\times {10}^5\text{A}$ for $i$ in equation (3).

    $\begin{array}{c}U=\frac{1}{2}\left(394.78\text{H}\right){\left(1.5\times {10}^5\text{A}\right)}^2\\ =4.4\times {10}^{12}\text{J}\end{array}$

Thus, the energy is $4.4\times {10}^{12}\text{J}$.

(c)

To determine

The time for supply the energy to the power plant.

Answer to Problem 38E

The time required to supply the power is $4.4\times {10}^3\text{s}$.

Explanation of Solution

Write the expression for the power.

    $U=Pt$.

Here, $U$ is the energy, $P$ is the power and $t$ is the time.

Rearrange the above equation to find the required time.

    $t=\frac{U}{P}$        (4)

Conclusion:

Substitute $4.4\times {10}^{12}\text{J}$ for $U$ and ${10}^9\text{W}$ for $P$ in equation (4).

    $\begin{array}{c}t=\frac{4.4\times {10}^{12}\text{J}}{{10}^9\text{W}}\\ =4.4\times {10}^3\text{s}\end{array}$

Thus, the time required to supply the power is $4.4\times {10}^3\text{s}$.