Question

Coil 1 has${\mathbf{L}}_{\mathbf{1}}\mathbf{=}\mathbf{25}\mathbf{}\mathbf{mH}$and ${\mathit{N}}_{\mathbf{1}}\mathbf{=}\mathbf{100}\mathbf{turns}$. Coil 2 has ${\mathit{L}}_{\mathbf{2}}\mathbf{=}\mathbf{40}\mathbf{}\mathbf{mH}$and ${\mathit{N}}_{\mathbf{2}}\mathbf{=}\mathbf{200}\mathbf{}\mathbf{turns}$. The coils are fixed in place; their mutual inductance M is $\mathit{M}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mH}$. A 6.0mAcurrent in coil 1 is changing at the rate of 4.0 A/s. (a) What magnetic flux ${\mathit{\phi }}_{\mathbf{12}}$links coil 1? (b) what self-induced emf appears in that coil? (c) What magnetic flux ${\mathit{\phi }}_{\mathbf{21}}$links coil 2? (d) what mutually induced emf appears in that coil?

Short answer

a)${\varphi }_{12}=1.5\times {10}^{-6}Wb$

b)$\epsilon =0.1V$

c)${\varphi }_{21}=9\times {10}^{-8}Wb$

d)$\epsilon =0.012V$

Step-by-step solution

Given

$$\begin{gathered} L_1=25\mathrm{mH}=0.025\mathrm{H} \\ {\mathrm{N}}_1=100 \\ {\mathrm{L}}_2=40\mathrm{mH}=0.040\mathrm{H} \\ {\mathrm{N}}_2=200 \\ \mathrm{M}=3.0\mathrm{mH}=0.003\mathrm{H} \\ \mathrm{I}=6.0\mathrm{mA}=0.006\mathrm{A} \\ \mathrm{di}/\mathrm{dt}=4.0\mathrm{A}/\mathrm{s} \end{gathered}$$

Understanding the concept

We have the formula for the magnetic flux induced in the coil, the self-induced emf and the mutually induced emf. So, using these equations and the given data, we can solve this problem.

Formula:

$$\begin{gathered} \varphi =\frac{LI}{N} \\ \epsilon =L\frac{di}{dt} \\ \varphi =\frac{MI}{N} \\ \epsilon =M\frac{di}{dt} \end{gathered}$$

(a) Calculate magnetic flux φ12 

We have,

$\varphi =\frac{LI}{N}$

Magnetic flux ${\varphi }_{12}$linking to coil 1 will be

$$\begin{gathered} {\varphi }_{12}=\frac{L_{1}I}{N_1} \\ {\varphi }_{12}=\frac{0.025\times 0.006}{100} \\ {\varphi }_{12}=1.5\times {10}^{-6}Wb \end{gathered}$$

(b) Calculate self-induced emf appeared in the coil 

We have the self – induced emf

$$\begin{gathered} \epsilon =L\frac{di}{dt} \\ \epsilon =0.025\times 4.0 \\ \epsilon =0.1V \end{gathered}$$

(c) Calculate magnetic flux φ21 

The flux can be written in terms of the mutual inductance, the current and the number of turns.

$\varphi =\frac{MI}{N}$

Magnetic flux ${\varphi }_{21}$linking to coil 2, will be denoted as${\varphi }_{21}$.

$$\begin{gathered} {\varphi }_{21}=\frac{0.003\times 0.006}{200} \\ {\varphi }_{21}=9\times {10}^{-8}Wb \end{gathered}$$

(d) Calculate mutually induced emf appears in the coil 

EMF induced can be written in terms of the mutual inductance and the rate of change of the current.

$$\begin{gathered} \epsilon =M\frac{di}{dt} \\ \epsilon =0.003\times 4.0 \\ \epsilon =0.012V \end{gathered}$$