Two circular coils of radii 20 cm and 30cm having number of turns 50 and 100 made of same material are connected in series. The ratio of the magnetic field of induction at their centres is

Options

A. `3:2`
B. `9:4`
C. `3:4`
D. `1:3`

Answer

To solve the problem, we need to find the ratio of the magnetic field induction at the centers of two circular coils connected in series. We will use the formula for the magnetic field at the center of a circular coil.

### Step-by-Step Solution:

1. **Identify the formula for the magnetic field at the center of a circular coil:**
The magnetic field \( B \) at the center of a circular coil is given by the formula:
\[ B = \frac{n \mu_0 I}{2r} \]
where:
- \( n \) = number of turns
- \( \mu_0 \) = permeability of free space (a constant)
- \( I \) = current through the coil
- \( r \) = radius of the coil

2. **Determine the ratio of the magnetic fields for the two coils:**
Since both coils are connected in series, the current \( I \) is the same for both coils. Therefore, we can express the ratio of the magnetic fields \( B_1 \) and \( B_2 \) as:
\[ \frac{B_1}{B_2} = \frac{n_1}{r_1} \cdot \frac{r_2}{n_2} \]
where:
- \( n_1 = 50 \) (number of turns for the first coil)
- \( n_2 = 100 \) (number of turns for the second coil)
- \( r_1 = 20 \, \text{cm} = 0.20 \, \text{m} \)
- \( r_2 = 30 \, \text{cm} = 0.30 \, \text{m} \)

3. **Substitute the values into the ratio:**
\[ \frac{B_1}{B_2} = \frac{50}{0.20} \cdot \frac{0.30}{100} \]

4. **Calculate the values:**
- First, calculate \( \frac{50}{0.20} = 250 \)
- Next, calculate \( \frac{0.30}{100} = 0.003 \)

Now, substitute these values back into the ratio:
\[ \frac{B_1}{B_2} = 250 \cdot 0.003 = 0.75 \]

5. **Express the ratio in simplest form:**
The ratio \( 0.75 \) can be expressed as:
\[ \frac{B_1}{B_2} = \frac{3}{4} \]

6. **Final Answer:**
Therefore, the ratio of the magnetic field induction at their centers is:
\[ \text{Ratio} = 3:4 \]