Introduction:
In this problem, we are given two concentric circular coils with different radii and opposite currents flowing through them. We need to calculate the magnetic field at the center of the coils.

Given:
- Number of turns in each coil (N1 = N2 = 10)
- Radii of the coils (r1 = 20 cm, r2 = 40 cm)
- Currents flowing through the coils (I1 = 0.2 A, I2 = 0.3 A)

Formula:
The magnetic field at the center of a circular coil is given by the formula:

B = (μ₀ * N * I) / (2 * R)

Where,
B = Magnetic field at the center of the coil
μ₀ = Permeability of free space (4π × 10^-7 T·m/A)
N = Number of turns in the coil
I = Current flowing through the coil
R = Radius of the coil

Calculation:
Coil 1:
- Number of turns (N1) = 10
- Radius (R1) = 20 cm = 0.2 m
- Current (I1) = 0.2 A

Using the formula, we can calculate the magnetic field at the center of coil 1:

B1 = (μ₀ * N1 * I1) / (2 * R1)

Coil 2:
- Number of turns (N2) = 10
- Radius (R2) = 40 cm = 0.4 m
- Current (I2) = 0.3 A

Using the formula, we can calculate the magnetic field at the center of coil 2:

B2 = (μ₀ * N2 * I2) / (2 * R2)

Net Magnetic Field:
Since the coils are concentric and carry currents in opposite directions, the net magnetic field at the center will be the difference between the magnetic fields of the two coils:

B_net = B2 - B1

Substitution:
- Permeability of free space (μ₀) = 4π × 10^-7 T·m/A
- Number of turns (N1) = N2 = 10
- Radius of coil 1 (R1) = 0.2 m
- Radius of coil 2 (R2) = 0.4 m
- Current in coil 1 (I1) = 0.2 A
- Current in coil 2 (I2) = 0.3 A

Final Calculation:
Substituting the given values into the formulas, we can calculate the magnetic field at the center:

B1 = (4π × 10^-7 T·m/A * 10 * 0.2 A) / (2 * 0.2 m)

B2 = (4π × 10^-7 T·m/A * 10 * 0.3 A) / (2 * 0.4 m)

B_net = B2 - B1

Final Answer:
The magnetic field at the center of the concentric coils is B_net Tesla.