A magnetic circuit of cross-sectional area consists of one part 3 cm long, of materia
relative permeability 1200 , and a second part 2 cm long of material having relative permeabi
With a 100 turn coil carrying 2 A , find the value of flux existing in the circuit.

Ask by Patel Mullins. in Ghana

Mar 28,2025

Question

A magnetic circuit of cross-sectional area consists of one part 3 cm long, of materia
relative permeability 1200 , and a second part 2 cm long of material having relative permeabi
With a 100 turn coil carrying 2 A , find the value of flux existing in the circuit.

Ask by Patel Mullins. in Ghana

Mar 28,2025

UpStudy AI · Accepted Answer

We first note that the magnetic flux in the circuit is given by

$$
\Phi=\frac{\mathcal{F}}{\mathcal{R}_{\text{total}}},
$$

where the magnetomotive force is

$$
\mathcal{F}=N\,I,
$$

and the total reluctance is the sum of the reluctances of the two series portions. For a section of magnetic path the reluctance is

$$
\mathcal{R}=\frac{l}{\mu_{0}\,\mu_{r}\,A}.
$$

In this problem the coil has

$$
N=100,\qquad I=2\text{ A}\quad \Longrightarrow\quad \mathcal{F}=100\times2=200\text{ A-turns}.
$$

The cross‐sectional area is given as

$$
A=0.4\,\text{cm}^{2}.
$$

Since

$$
1\,\text{cm}^{2}=10^{-4}\,\text{m}^{2},\qquad A=0.4\times10^{-4} = 4.0\times10^{-5}\,\text{m}^{2}.
$$

We are told the circuit consists of two parts:
1. A 3 cm long section ($$l_{1}=3\,\text{cm}=0.03\,\text{m}$$) of material with relative permeability $$\mu_{r1}=1200$$.
2. A 2 cm long section ($$l_{2}=2\,\text{cm}=0.02\,\text{m}$$) of material with relative permeability $$\mu_{r2}=200$$.

(The second permeability is assumed to be 200; if a different value were intended, the procedure would be analogous.)

The permeability of free space is

$$
\mu_{0}=4\pi \times10^{-7}\,\text{H/m}.
$$

We now calculate the reluctance for each section.

**Reluctance of the first section:**

$$
\mathcal{R}_{1}=\frac{l_{1}}{\mu_{0}\,\mu_{r1}\,A}=\frac{0.03}{(4\pi\times10^{-7})(1200)(4.0\times10^{-5})}.
$$

Let us compute the denominator step by step:

1. Compute $$\mu_{0}\,\mu_{r1}$$:

$$
4\pi\times10^{-7}\times1200=4800\pi\times10^{-7}.
$$

2. Multiply by $$A$$:

$$
4800\pi\times10^{-7}\times4.0\times10^{-5}=4800\times4.0\pi\times10^{-12}=19200\pi\times10^{-12}=1.92\pi\times10^{-8}.
$$

Thus,

$$
\mathcal{R}_{1}=\frac{0.03}{1.92\pi\times10^{-8}}.
$$

Using $$\pi\approx3.1416$$, we have

$$
1.92\pi\approx1.92\times3.1416\approx6.035.
$$

Then

$$
\mathcal{R}_{1}\approx\frac{0.03}{6.035\times10^{-8}}\approx4.97\times10^{5}\,\text{A/Wb}.
$$

**Reluctance of the second section:**

$$
\mathcal{R}_{2}=\frac{l_{2}}{\mu_{0}\,\mu_{r2}\,A}=\frac{0.02}{(4\pi\times10^{-7})(200)(4.0\times10^{-5})}.
$$

Compute the denominator:

1. Compute $$\mu_{0}\,\mu_{r2}$$:

$$
4\pi\times10^{-7}\times200=800\pi\times10^{-7}.
$$

2. Multiply by $$A$$:

$$
800\pi\times10^{-7}\times4.0\times10^{-5}=800\times4.0\pi\times10^{-12}=3200\pi\times10^{-12}=3.2\pi\times10^{-9}.
$$

Thus,

$$
\mathcal{R}_{2}=\frac{0.02}{3.2\pi\times10^{-9}}.
$$

Again, using $$\pi\approx3.1416$$,

$$
3.2\pi\approx3.2\times3.1416\approx10.053.
$$

Then

$$
\mathcal{R}_{2}\approx\frac{0.02}{10.053\times10^{-9}}\approx1.99\times10^{6}\,\text{A/Wb}.
$$

**Total reluctance:**

$$
\mathcal{R}_{\text{total}}=\mathcal{R}_{1}+\mathcal{R}_{2}\approx(4.97\times10^{5}+1.99\times10^{6})\,\text{A/Wb}\approx2.487\times10^{6}\,\text{A/Wb}.
$$

**Magnetic flux:**

Finally, using

$$
\Phi=\frac{\mathcal{F}}{\mathcal{R}_{\text{total}}}=\frac{200}{2.487\times10^{6}},
$$

we obtain

$$
\Phi\approx8.04\times10^{-5}\,\text{Wb}.
$$

Thus, the magnetic flux in the circuit is approximately

$$
\boxed{8.0\times10^{-5}\,\text{Weber}}.
$$
The magnetic flux in the circuit is approximately $$8.0 \times 10^{-5}$$ Weber.

A magnetic circuit of cross-sectional area consists of one part 3 cm long, of materia
relative permeability 1200 , and a second part 2 cm long of material having relative permeabi
With a 100 turn coil carrying 2 A , find the value of flux existing in the circuit.

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A magnetic circuit of cross-sectional area consists of one part 3 cm long, of materia relative permeability 1200 , and a second part 2 cm long of material having relative permeabi With a 100 turn coil carrying 2 A , find the value of flux existing in the circuit.

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Latest Engineering Questions

A magnetic circuit of cross-sectional area consists of one part 3 cm long, of materia
relative permeability 1200 , and a second part 2 cm long of material having relative permeabi
With a 100 turn coil carrying 2 A , find the value of flux existing in the circuit.