To double the torque acting on a coil of n-turns, when placed in a magnetic field

Options

A. area of the coil and the magnetic induction should be doubled
B. area and current through the coil should be doubled
C. only area of coil should be doubled
D. number of turns are to be halved

Answer

To solve the problem of how to double the torque acting on a coil of n turns when placed in a magnetic field, we start with the formula for torque (τ) acting on a coil:

### Step 1: Understand the formula for torque
The torque (τ) acting on a coil in a magnetic field is given by the formula:
\[ \tau = n \times i \times A \times B \times \sin(\theta) \]
where:
- \( n \) = number of turns in the coil
- \( i \) = current flowing through the coil
- \( A \) = area of the coil
- \( B \) = magnetic field strength
- \( \theta \) = angle between the magnetic field and the normal to the coil

### Step 2: Determine the condition for doubling the torque
To double the torque, we need to find a way to modify the parameters in the torque formula such that:
\[ \tau' = 2\tau \]
where \( \tau' \) is the new torque after modifications.

### Step 3: Analyze the options
We will analyze each option provided in the question to see which one leads to doubling the torque.

#### Option 1: Area of the coil and magnetic induction should be doubled
If we double both the area (A) and magnetic field (B), the new torque becomes:
\[ \tau' = n \times i \times (2A) \times (2B) \times \sin(\theta) = 4n \times i \times A \times B \times \sin(\theta) = 4\tau \]
This results in quadrupling the torque, so this option is incorrect.

#### Option 2: Area and current through the coil should be doubled
If we double both the area (A) and current (i), the new torque becomes:
\[ \tau' = n \times (2i) \times (2A) \times B \times \sin(\theta) = 4n \times i \times A \times B \times \sin(\theta) = 4\tau \]
Again, this results in quadrupling the torque, so this option is also incorrect.

#### Option 3: Only area of the coil should be doubled
If we only double the area (A), the new torque becomes:
\[ \tau' = n \times i \times (2A) \times B \times \sin(\theta) = 2n \times i \times A \times B \times \sin(\theta) = 2\tau \]
This results in exactly doubling the torque, so this option is correct.

#### Option 4: Number of turns are to be halved
If we halve the number of turns (n), the new torque becomes:
\[ \tau' = \left(\frac{n}{2}\right) \times i \times A \times B \times \sin(\theta) = \frac{1}{2}n \times i \times A \times B \times \sin(\theta) = \frac{1}{2}\tau \]
This results in halving the torque, so this option is incorrect.

### Conclusion
The only option that correctly doubles the torque is:
**Option 3: Only area of the coil should be doubled.**

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