Two similar coils of radius `R` are lying concentriclaly with their planes at right angels to each other. The currents flowing in them are `I` and `2I` respectively. The resulant magntic field induction at the centre will be

Options

A. `(sqrt(5mu_(0)I))/(2R)`
B. `(3mu_(0)I)/(2R)`
C. `(mu_(0)I)/(2R)`
D. `(mu_(0)i)/(R)`

Answer

To find the resultant magnetic field induction at the center of two concentric coils with currents flowing in them, we can follow these steps:

### Step 1: Understand the magnetic field produced by a single coil
The magnetic field \( B \) at the center of a single circular coil of radius \( R \) carrying a current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Calculate the magnetic field due to the first coil
For the first coil with current \( I \):
\[ B_1 = \frac{\mu_0 I}{2R} \]

### Step 3: Calculate the magnetic field due to the second coil
For the second coil with current \( 2I \):
\[ B_2 = \frac{\mu_0 (2I)}{2R} = \frac{\mu_0 I}{R} \]

### Step 4: Determine the direction of the magnetic fields
Since the coils are oriented at right angles to each other, the magnetic fields \( B_1 \) and \( B_2 \) will also be perpendicular to each other.

### Step 5: Use the Pythagorean theorem to find the resultant magnetic field
The resultant magnetic field \( B_{net} \) can be found using the Pythagorean theorem:
\[ B_{net} = \sqrt{B_1^2 + B_2^2} \]

### Step 6: Substitute the values of \( B_1 \) and \( B_2 \)
Substituting the expressions for \( B_1 \) and \( B_2 \):
\[ B_{net} = \sqrt{\left(\frac{\mu_0 I}{2R}\right)^2 + \left(\frac{\mu_0 I}{R}\right)^2} \]

### Step 7: Simplify the expression
Calculating the squares:
\[ B_{net} = \sqrt{\left(\frac{\mu_0^2 I^2}{4R^2}\right) + \left(\frac{\mu_0^2 I^2}{R^2}\right)} \]
\[ = \sqrt{\frac{\mu_0^2 I^2}{4R^2} + \frac{4\mu_0^2 I^2}{4R^2}} = \sqrt{\frac{5\mu_0^2 I^2}{4R^2}} \]

### Step 8: Factor out the common terms
\[ B_{net} = \frac{\mu_0 I}{2R} \sqrt{5} \]

### Final Result
Thus, the resultant magnetic field induction at the center of the coils is:
\[ B_{net} = \frac{\mu_0 I \sqrt{5}}{2R} \]