Q.A solenoid has a core of a material with relative permeability $400$. The windings of the solenoid are insulated from the core and carry a current of $2\ \text{A}$. If the number of turns is $1000$ per metre, calculate

Concept: Magnetic Materials – Magnetization. The magnetic field $H$ is due to the free current in the solenoid; the material’s response adds magnetization $M$, giving the total field $B = \mu_0 (H + M)$. Relative permeability $\mu_r$ relates $B$ and $H$ via $B = \mu_r \mu_0 H$.

Step 1 – Magnetizing field $H$

For an ideal solenoid, $H = n I$, where $n = 1000$ turns/m and $I = 2\ \text{A}$.

Step 2 – Magnetic flux density $B$

Using $B = \mu_r \mu_0 H$ with $\mu_r = 400$ and $\mu_0 = 4\pi \times 10^{-7}\ \text{H/m}$:

Step 3 – Magnetization $M$

From $B = \mu_0 (H + M)$, we get $M = \frac{B}{\mu_0} - H$. …

The core of this problem is understanding the three magnetic quantities — $H$, $M$, and $B$ — and how they relate inside a material. In a solenoid, the field produced by the free current alone is $H$. The material responds by developing a magnetization $M$, which adds to $H$ to give the total magnetic field $B$. The relative permeability $\mu_r$ tells us how strongly the material amplifies the field.

Let’s work through each part step by step.

1. Find $H$ — the magnetic field intensity For a long solenoid, $H$ depends only on the free current and the number of turns per metre, not on the core material.

where $n = 1000\ \text{turns/m}$ and $I = 2\ \text{A}$.

This is the field that would exist in vacuum if the core were absent.

2. Find $B$ — the magnetic flux density Inside a linear magnetic material, $B$ is related to $H$ by:

Given $\mu_r = 400$ and $\mu_0 = 4\pi \times 10^{-7}\ \text{H/m}$:

3. Find $M$ — the magnetization Magnetization $M$ is the magnetic moment per unit volume of the core material. The fundamental relation is:

Rearranging:

Substitute $B$ from step 2:

First term: $\frac{1.0053}{1.2566 \times 10^{-6}} \approx 8.00 \times 10^5\ \text{A/m}$

So:

Alternatively, using $\mu_r$ directly: …

Method: Magnetization Relations in a Solenoid Core

We use the magnetic circuit approach with definitions of magnetizing field ($H$), magnetization ($M$), and magnetic flux density ($B$).

Step 1: Calculate $H$ (Magnetizing Field)

$H$ depends only on the free current in the solenoid windings — not on the core material.

where:

• $n = 1000$ turns/m
• $I = 2\ \text{A}$

Step 2: Calculate $B$ (Magnetic Flux Density)

Using the relation with relative permeability $\mu_r = 400$:

where $\mu_0 = 4\pi \times 10^{-7}\ \text{H/m}$

Step 3: Calculate $M$ (Magnetization)

From the fundamental relation:

Rearrange:

Alternatively, using $M = (\mu_r - 1)H$:

Step 4: Calculate $I_m$ (Magnetising Current)

Magnetising current is the equivalent current that would produce the same $B$ if the core were absent. It is related to $M$ by:

$I_m = M \times \text{length per turn?}$ …

Here are the common mistakes students make on this problem, along with how to avoid each.

1. Confusing $H$ with $B$

Mistake:

Students often plug $\mu_r$ into the formula for $H$, writing $H = \mu_r n I$.

Why it’s wrong:

$H$ (magnetizing field) depends only on the free current and geometry — not on the core material.

The formula is:

where $n = 1000$ turns/m and $I = 2\ \text{A}$.

How to avoid:

Remember: $H$ is the field due to free currents alone. The core’s response comes later, in $B$ and $M$.

2. Forgetting to convert units or misreading $n$

Mistake:

Using $n = 1000$ without checking units, or writing $n = 1000\ \text{turns}$ instead of $1000\ \text{turns/m}$.

How to avoid:

Always write units explicitly. Here $n = 1000\ \text{m}^{-1}$ is already given. If it were “1000 turns per metre,” that’s exactly $n = 1000$.

3. Using $B = \mu_0 H$ for a magnetic core

Mistake:

Writing $B = \mu_0 H$ even when a core is present.

Why it’s wrong:

Inside a material, $B = \mu_0 \mu_r H$. For air, $\mu_r = 1$, but here $\mu_r = 400$.

Correct formula:

How to avoid:

Always check: is there a core? If yes, use $\mu = \mu_0 \mu_r$.

4. Mixing up $M$ and $B$ or $M$ and $H$

Mistake:

Writing $M = \chi_m H$ but forgetting $\chi_m = \mu_r - 1$, or writing $M = \mu_r H$.

Correct relation:

How to avoid:

Memorize the chain:

$\mu_r \rightarrow \chi_m = \mu_r - 1 \rightarrow M = \chi_m H$

5. Sign errors in the magnetizing current $I_m$

Mistake:

Writing $I_m = (\mu_r - 1) I$ without considering $n$.

Why it’s wrong:

Magnetizing current is defined as the equivalent current that would produce the same $B$ if the core were absent. The correct formula is:

Standard definition:

where $I$ is the free current. However, some textbooks define it as:

How to avoid:

Check your textbook’s definition. In most Indian board exams (CBSE, NCERT pattern), the magnetizing current per unit length is:

$I_m = (\mu_r - 1) I$ …