Two concentric circular wire loops in the same plane each carry a current. The larger loop has a current of 8.62 A circulating clockwise and has a radius of 6.09 cm. The smaller loop has a radius of 4.42 cm. What is the current in the smaller loop if the total magnetic field at the center of the system is zero?
B1=B2
μ₀•I1/2•R = μ₀•I2/2•r
I2=I1•r/R=8.62•4.42/6.09 = 6.26 A (anticlockwise)
To find the current in the smaller loop, we can use Ampere's Law. According to Ampere's Law, the magnetic field at the center of the system due to a circular loop is given by:
B = (μ₀ * I) / (2 * R)
Where B is the magnetic field, μ₀ is the permeability of free space (μ₀ = 4π × 10^(-7) T·m/A), I is the current, and R is the radius of the loop.
Since the total magnetic field at the center is zero, we can set up two equations based on the formula above:
For the larger loop:
B₁ = (μ₀ * I₁) / (2 * R₁)
For the smaller loop:
B₂ = (μ₀ * I₂) / (2 * R₂)
Since the two loops are concentric and in the same plane, the magnetic field at the center is the vector sum of the magnetic fields due to each loop. Since the total magnetic field is zero, we have:
B = B₁ + B₂
Substituting the equations for B₁ and B₂, we have:
(μ₀ * I₁) / (2 * R₁) + (μ₀ * I₂) / (2 * R₂) = 0
Simplifying the equation, we get:
I₁ / R₁ + I₂ / R₂ = 0
Now we can solve for I₂, the current in the smaller loop. Rearranging the equation, we have:
I₂ / R₂ = -I₁ / R₁
Cross-multiplying, we get:
I₂ * R₁ = -I₁ * R₂
Finally, solving for I₂, we have:
I₂ = (-I₁ * R₂) / R₁
Now we can plug in the values:
I₁ = 8.62 A
R₁ = 6.09 cm = 0.0609 m
R₂ = 4.42 cm = 0.0442 m
Substituting these values into the equation, we get:
I₂ = (-8.62 A * 0.0442 m) / 0.0609 m
Calculating this, we find that the current in the smaller loop is approximately -6.25 A (negative because it must have the opposite direction to cancel out the magnetic field of the larger loop). Therefore, the current in the smaller loop is 6.25 A clockwise.