Question

The ratio of the magnetic fields produced at the centre of a solenoid for a flow of current 1 A to that produced inside toroid for the flow of current 2 A both having same number of turns per unit length is

• A. 1 : 1
• B. 1 : 2
• C. 2 : 1
• D. 1 : 4
• E. 4 : 1

Hint:

When turns per unit length are same, magnetic field depends only on current → directly proportional.

Answer 1

The Correct Option is B

Concept: Magnetic field in current-carrying conductors depends on geometry:
• For a long solenoid: \[ B = \mu_0 n I \]
• For a toroid: \[ B = \mu_0 n I \] (where \(n\) is number of turns per unit length along circular path) Thus, in both cases, magnetic field is directly proportional to current when \(n\) is same.

Step 1: Understand given conditions carefully.
We are given:
• Current in solenoid \(I_1 = 1A\)
• Current in toroid \(I_2 = 2A\)
• Number of turns per unit length is same in both cases

Step 2: Write magnetic field expression for solenoid.
For a long solenoid: \[ B_{\text{solenoid}} = \mu_0 n I_1 \] Substituting: \[ B_1 = \mu_0 n \times 1 = \mu_0 n \]

Step 3: Write magnetic field expression for toroid.
For a toroid: \[ B_{\text{toroid}} = \mu_0 n I_2 \] Substituting: \[ B_2 = \mu_0 n \times 2 = 2\mu_0 n \]

Step 4: Compare both magnetic fields.
\[ \frac{B_1}{B_2} = \frac{\mu_0 n}{2\mu_0 n} \]

Step 5: Cancel common terms.
\[ \frac{B_1}{B_2} = \frac{1}{2} \]

Step 6: Write final ratio.
\[ B_1 : B_2 = 1 : 2 \]

Step 7: Final conclusion. \[ \boxed{1 : 2} \]