Q.Consider a tightly wound $100$ turn coil of radius $10\ \text{cm}$, carrying a current of $1\ \text{A}$. What is the magnitude of the magnetic field at the centre of the coil?

The key idea is the magnetic field at the centre of a circular current-carrying coil, given by the formula derived from the Biot–Savart law.

For a single circular loop of radius $R$ carrying current $I$, the field at the centre is $B = \frac{\mu_0 I}{2R}$. For a coil with $N$ tightly wound turns, the fields of each turn add directly, so:

Here, $N = 100$, $I = 1\ \text{A}$, $R = 10\ \text{cm} = 0.1\ \text{m}$, and $\mu_0 = 4\pi \times 10^{-7}\ \text{T m/A}$.

Substitute: …

Why the Biot–Savart Law?

The magnetic field at the centre of a current-carrying circular loop arises from the Biot–Savart law. Each tiny current element $I\,d\vec{l}$ on the loop produces a magnetic field at the centre that points along the axis (perpendicular to the plane of the loop). Because of symmetry, the contributions from all elements add constructively -- there is no cancellation. The key insight: every element is at the same distance $R$ from the centre, and the angle between $d\vec{l}$ and the radial vector is always $90^\circ$, so the cross product simplifies beautifully.

This is the central result for a tightly wound coil of $N$ turns. Let's derive it step by step.

Step-by-step solution

1. Start with a single turn. For a single circular loop of radius $R$, carrying current $I$, the magnetic field at the centre is

This comes from integrating the Biot–Savart law: each element $I\,dl$ contributes $dB = \frac{\mu_0}{4\pi} \frac{I\,dl \sin\theta}{r^2}$. Here $r = R$ and $\theta = 90^\circ$ (since $d\vec{l}$ is tangent and the vector from element to centre is radial), so $\sin\theta = 1$. The integral $\oint dl = 2\pi R$ gives the result.

2. Account for multiple turns. The coil has $N = 100$ turns, all tightly wound so they essentially occupy the same radius $R$. The fields from each turn add linearly (superposition). Hence

3. Plug in the numbers.

   • $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$
   • $N = 100$
   • $I = 1\ \text{A}$
   • $R = 10\ \text{cm} = 0.1\ \text{m}$

   So …

Method: Magnetic Field at the Centre of a Circular Coil (Biot–Savart Law)

This is a standard application of the Biot–Savart law for a circular current loop. The symmetry of the problem allows a direct formula.

Steps

1. Recall the formula For a single circular loop of radius $R$ carrying current $I$, the magnetic field at the centre is:

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

2. Account for multiple turns For a tightly wound coil of $N$ turns, each turn contributes the same field at the centre (since they are concentric). The total field is simply:

3. Substitute the given values
   • $N = 100$
   • $I = 1\ \text{A}$
   • $R = 10\ \text{cm} = 0.1\ \text{m}$

4. Simplify step-by-step
   • Denominator: $2 \times 0.1 = 0.2$
   • Numerator: $4\pi \times 10^{-7}$
   • So: …

Here are the most common mistakes students make when solving this problem, along with how to avoid each one.

1. Forgetting the Number of Turns ($N$)

The Mistake:

Students often use the formula for a single loop ($B = \frac{\mu_0 I}{2R}$) and forget to multiply by the number of turns $N$.

Why it happens:

The problem says "tightly wound 100 turn coil." A coil is not a single loop — each turn contributes to the field at the centre.

How to Avoid:

Always check: Is it a single loop or a coil with $N$ turns?

For a coil, the correct formula is:

Here, $N = 100$, so the field is 100 times that of a single loop.

2. Using the Wrong Radius Units

The Mistake:

Plugging $R = 10$ directly into the formula without converting to metres.

Why it happens:

The radius is given in cm, but the SI unit for $R$ in the formula is metres.

How to Avoid:

Always convert cm to m before substituting:

If you forget, your answer will be off by a factor of 10.

3. Confusing the Formula for a Solenoid vs. a Circular Coil

The Mistake:

Using $B = \mu_0 n I$ (the solenoid formula) instead of the circular coil formula.

Why it happens:

Both involve "coils" and "turns," but the geometry is different. A solenoid is long and straight; a circular coil is a flat loop.

How to Avoid:

• Circular coil (flat): $B = \frac{\mu_0 N I}{2R}$ (centre field)
• Solenoid (long): $B = \mu_0 n I$ (inside field, $n$ = turns per unit length)

If the problem says "radius" and "centre," it's a circular coil.

4. Forgetting the Direction of the Magnetic Field

The Mistake:

Only calculating magnitude but ignoring that the question asks for "magnitude" — some students still try to assign a direction incorrectly.

Why it happens:

Direction is important in many problems, but here only magnitude is asked. Overthinking can lead to errors.

How to Avoid: …