Given:
- Two circular loops of radius R are concentrically lying with their planes at right angles to each other.
- The current flowing in one loop is I, and in the other loop is I√3.

To find:
- The magnetic field at the center of the coil.

Explanation:
1. Magnetic Field Due to a Circular Loop:
The magnetic field at the center of a circular loop is given by the formula:

B = μ₀I / (2R)

Where:
- B is the magnetic field at the center of the loop.
- μ₀ is the permeability of free space (constant).
- I is the current flowing through the loop.
- R is the radius of the loop.

2. Magnetic Fields of the Two Loops:
Let's calculate the magnetic fields due to the two loops individually.

For the first loop:
- Current flowing through the loop = I
- Radius of the loop = R

Using the formula, the magnetic field at the center of the first loop is given by:

B₁ = μ₀I / (2R)

For the second loop:
- Current flowing through the loop = I√3
- Radius of the loop = R

Using the formula, the magnetic field at the center of the second loop is given by:

B₂ = μ₀(I√3) / (2R)

3. Magnetic Field at the Center of the Combined Loops:
Since the two loops are concentrically lying with their planes at right angles to each other, the magnetic fields they produce at their centers are perpendicular to each other. Therefore, we can use the Pythagorean theorem to find the resultant magnetic field.

Using the Pythagorean theorem, the resultant magnetic field at the center of the combined loops is given by:

B = √(B₁² + B₂²)

Substituting the values of B₁ and B₂, we get:

B = √((μ₀I / (2R))² + (μ₀(I√3) / (2R))²)

Simplifying the equation further, we get:

B = μ₀I√(1/4 + 3/4)
B = μ₀I√1
B = μ₀I

Therefore, the magnetic field at the center of the coil is equal to μ₀I.

Conclusion:
The magnetic field at the center of the coil is equal to μ₀I, where I is the current flowing through the loop.