Q.A closely wound solenoid $80\ \text{cm}$ long has $5$ layers of windings of $400$ turns each. The diameter of the solenoid is $1.8\ \text{cm}$. If the current carried is $8.0\ \text{A}$, estimate the magnitude of $B$ inside the solenoid near its centre.

Concept: Magnetic Field of a Finite Solenoid — For a long solenoid, the field near the centre is nearly uniform and given by $B = \mu_0 n I$, where $n$ is the total number of turns per unit length.

Step 1: Find total turns.

$N = 5 \times 400 = 2000$ turns.

Step 2: Find turns per unit length.

Length $L = 80\ \text{cm} = 0.80\ \text{m}$.

$n = \frac{N}{L} = \frac{2000}{0.80} = 2500\ \text{turns/m}$.

Step 3: Apply formula.

$B = \mu_0 n I = (4\pi \times 10^{-7}) \times 2500 \times 8.0$. …

The key insight here is that a solenoid’s magnetic field near its centre depends only on the total number of turns per unit length and the current — not on the number of layers or the wire diameter, as long as the solenoid is long compared to its radius. The field lines are nearly parallel and uniform inside, so we can use the ideal solenoid formula directly.

Let’s work through it step by step.

1. Find the total number of turns.

   The solenoid has 5 layers, each with 400 turns.

   Total turns $N = 5 \times 400 = 2000$ turns.

2. Find the length of the solenoid in metres.

   Length $L = 80\ \text{cm} = 0.80\ \text{m}$.

3. Calculate the number of turns per unit length $n$.

4. Apply the formula for the magnetic field inside a long solenoid. For an ideal solenoid (length >> radius), the field near the centre is:

where $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$ and $I = 8.0\ \text{A}$.

5. Plug in the numbers.

First, $4\pi \times 10^{-7} \approx 1.2566 \times 10^{-6}$.

Then $2500 \times 8.0 = 20000$.

So $B \approx 1.2566 \times 10^{-6} \times 20000 = 2.5132 \times 10^{-2}\ \text{T}$.

Rounding to two significant figures (since the given data has two significant figures in current and length), we get: …

Method: Ideal Solenoid Approximation (Ampere's Law)

This method uses the fact that for a long, tightly wound solenoid, the magnetic field inside is nearly uniform and directed along the axis.

Steps

1. Find the total number of turns The solenoid has $5$ layers, each with $400$ turns.

2. Calculate the number of turns per unit length Length of solenoid, $L = 80\ \text{cm} = 0.80\ \text{m}$

3. Apply the formula for $B$ inside an ideal solenoid From Ampere's law, for a long solenoid:

where $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$ and $I = 8.0\ \text{A}$.

4. Substitute and compute

$B = 4\pi \times 10^{-7} \times 20000$ …

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

1. Mistaking the total number of turns

The error:

Students often take $N = 400$ (the turns per layer) instead of the total turns across all layers.

Why it happens:

The phrase “5 layers of windings of 400 turns each” is misread as “400 turns total.”

How to avoid:

Always multiply:

Write it down explicitly before plugging into any formula.

2. Using the wrong length unit

The error:

Plugging $L = 80$ directly into the formula without converting to metres.

Why it happens:

The formula $B = \mu_0 n I$ uses SI units — length must be in metres.

How to avoid:

Convert immediately:

Then compute turns per metre:

3. Confusing $n$ (turns per metre) with $N$ (total turns)

The error:

Using $B = \mu_0 N I$ instead of $B = \mu_0 n I$.

Why it happens:

Memorising the formula without understanding that $n$ is the density of turns.

How to avoid:

Remember:

Always compute $n = N/L$ first, then substitute.

4. Including the diameter in the calculation

The error:

Using the diameter $1.8\ \text{cm}$ to find area or radius, then trying to use a formula like $B = \frac{\mu_0 N I}{2R}$.

Why it happens:

Confusing a solenoid with a circular loop or a toroid.

How to avoid:

For an ideal solenoid (length >> diameter), the field inside is uniform and independent of the cross-sectional area. The diameter is a distractor here — ignore it unless the problem asks for flux.

5. Forgetting $\mu_0$ or using the wrong value

The error: …