Question

The formula for magnetic field of a certain component is given by \[ B = \frac{\mu_0 Ni}{2\pi r} \] where, \(N\) = total number of turns, \(i\) = current, \(r\) = radius. The component is:

• A. ring
• B. solenoid
• C. toroid
• D. None of these

Hint:

Toroid field depends on \( \frac{1}{r} \), unlike solenoid (constant field).

Answer 1

The Correct Option is C

The given formula for the magnetic field is:

\(B = \frac{\mu_0 Ni}{2\pi r}\),

where:

• \(\mu_0\) is the permeability of free space.
• \(N\) is the total number of turns.
• \(i\) is the current flowing through the coil.
• \(r\) is the radius.

The component described by this formula is a toroid. Let's understand why this is the correct answer:

1. **Toroid Characteristics:** A toroid is a doughnut-shaped coil used to create a magnetic field. The magnetic field inside a toroid is essentially circular and remains confined within the core, which is why the formula contains \(2\pi r\) in the denominator. This accounts for the circumference along which the magnetic field is spread.
2. **Formula Comparison:** In the formula \(B = \frac{\mu_0 Ni}{2\pi r}\), the presence of \(2\pi r\) indicates a circular or cylindrical path, typical in toroidal structures.
3. **Exclusion of Other Options:**
   • Ring: Typically, a ring without a coil does not produce any magnetic field due to current. The formula for magnetic fields involves coil characteristics with current.
   • Solenoid: The magnetic field inside a solenoid is given by \(B = \mu_0 \frac{Ni}{L}\), where \(L\) is the length of the solenoid. This formula is different from the one provided.
   • None of these: Not applicable as the provided formula specifically describes a toroid.

Therefore, the correct answer is a toroid.