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
To find the expression for the vector 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 enclosed by the loop and the permeability of free space (μ₀).
In this case, the loop consists of a circular section and two straight sections. Let's divide the loop into three parts:
1. Circular section of radius R:
The magnetic field at the center of this circular section is given by the Biot-Savart law, which states that the magnetic field at a point due to a small segment of current-carrying wire is proportional to the current, the element length, and inversely proportional to the square of the distance from the current element.
The magnitude of the magnetic field at the center of the circular section can be calculated as:
B₁ = (μ₀ I) / (2R)
Note that the direction of the magnetic field at every point on the circular section is perpendicular to the plane of the loop and directed outwards.
2. Straight section 1:
Since this section is straight and parallel to the magnetic field lines that will be generated by the circular section, the magnetic field at the center of this section will be zero.
3. Straight section 2:
Similar to straight section 1, the magnetic field at the center of this section will also be zero.
Since the magnetic field is a vector quantity, we need to consider the directions carefully. The magnetic field produced by the circular section is perpendicular to the plane of the loop and directed outwards. Therefore, the direction of the magnetic field at the center of the loop will be the same as that of the circular section.
So, the expression for the vector magnetic field at the center of the loop can be given as:
B = B₁ (in the outward direction)
where B₁ = (μ₀ I) / (2R) is the magnitude of the magnetic field at the center of the circular section.
To find the expression for the vector magnetic field at the center of the loop, we can use Ampere's law. Ampere's law states that the magnetic field around a closed loop is directly proportional to the current passing through the loop.
To apply Ampere's law to this situation, we can consider a circular Amperian loop of radius R that lies in the same plane as the conductor loop. The Amperian loop should enclose the conductor loop completely. Since the wire carries a current I, the current passing through the Amperian loop is also I.
Using Ampere's law, we have:
∮ B · dl = μ₀I,
where B is the magnetic field, dl is an infinitesimal vector element along the Amperian loop, ∮ represents the line integral around the loop, and μ₀ is the permeability of free space.
Since the magnetic field is constant along the straight sections of the conductor loop and zero outside the conductor loop, the only contribution to the line integral comes from the circular section of the conductor loop.
The magnetic field B is tangential to the circular conductor loop and has the same magnitude at all points on the circular section. Thus, we can take it out of the line integral:
B ∮ dl = μ₀I.
The line integral can be evaluated as the circumference of the circular Amperian loop:
B (2πR) = μ₀I.
Therefore, the expression for the magnetic field at the center of the loop is:
B = (μ₀I) / (2πR).
This expression gives the magnitude and direction of the magnetic field vector at the center of the loop.