A conductor consists of a circular loop of radius R and two straight, long sections The wire lies in the plane of the paper and carries a current I. Find an expression for the vector magnetic field at the center of the loop
You are so lucky, you can use superposition.
figure the B due to a long wire.
Figure the B at the center of the loop.
add them.
To find the expression for the magnetic field at the center of the loop, we can use Ampere's Law. Ampere's Law states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space.
Mathematically, Ampere's Law is given as:
∮ B · dl = μ₀ * I,
where B is the magnetic field, dl is a differential element of the path, μ₀ is the permeability of free space, and I is the current passing through the loop.
Now, let's consider a circular loop of radius R with a current I flowing through it. We want to find the magnetic field at the center of the loop. To apply Ampere's Law, we need to choose a closed loop that encloses the current and simplifies the calculation.
In this case, we can choose the circular loop itself as the closed loop. The path integral of the magnetic field around this loop will give us the desired result.
The circular loop consists of two symmetrical halves. Due to symmetry, the contribution to the magnetic field from one half cancels out the contribution from the other half at the center of the loop. Therefore, we only need to consider the magnetic field contribution from one half.
The magnetic field from a straight, long wire at a distance r from the wire is given by the Biot-Savart Law:
B = (μ₀ * I) / (2π * r),
where μ₀ is the permeability of free space, I is the current, and r is the distance from the wire.
Now, consider a differential element dl on the circular loop. Its contribution to the magnetic field at the center is given by the Biot-Savart Law. Assuming the differential element is located at a distance s from the center of the loop, the magnetic field contribution from this element is:
d𝐵 = (μ₀ * I * ds) / (2π * s),
where ds is the differential length element on the circular loop.
To find the total magnetic field at the center of the loop, we need to integrate the contributions from all the differential elements around the loop.
∮ B · dl = μ₀ * I,
∫(B · dl) = μ₀ * I,
∫(d𝐵 · dl) = μ₀ * I,
∫[(μ₀ * I * ds) / (2π * s)] · dl = μ₀ * I,
Since dl is along the circumference of the loop, we have dl = R · dθ,
∫[(μ₀ * I * ds) / (2π * s)] · (R · dθ) = μ₀ * I,
∫[(μ₀ * I * R * ds) / (2π * s)] · dθ = μ₀ * I,
∫[(μ₀ * I * R) / (2π * s)] · dθ = μ₀ * I,
Let's define λ = (μ₀ * I * R) / (2π), which represents the current per unit length of the circular loop.
∫[λ / s] · dθ = μ₀ * I,
λ * ∫(dθ / s) = μ₀ * I,
λ * ln(s) = μ₀ * I,
ln(s) = (μ₀ * I) / λ,
s = e^((μ₀ * I) / λ),
Substituting back the value of λ, we get:
s = e^((2π * s) / (μ₀ * I * R)).
Therefore, the expression for the magnetic field at the center of the loop is:
B = (μ₀ * I) / (2π * s) = (μ₀ * I) / (2π * e^((2π * s) / (μ₀ * I * R))).
This is the desired expression for the vector magnetic field at the center of the loop.