Question:
A long, straight wire carries a current of 4 A in the positive x-direction. A circular loop, with a radius of 0.1 m, lies in the y-z plane centered at the origin, O. The current in the wire creates a magnetic field in the region. Let's assume the magnetic field at any point inside the circular loop is given by the equation:
B = k⋅y
where B is the magnetic field, k is a constant, and y is a coordinate along the y-axis.
a) Calculate the constant k.
b) Calculate the value of the magnetic field at a point P, located at (0, 0.1, 0) m.
c) A small current-carrying loop with a current of 2 A is placed inside the circular loop. The small loop lies in the y-z plane and has a radius of 0.05 m. Calculate the magnitude and direction of the net magnetic force experienced by the small loop due to the current in the wire.
Answer:
a) In order to calculate the constant k, let's consider a point on the loop with coordinates (0, y, 0). The magnetic field at this point is given by B = k⋅y. Since the current in the wire creates a magnetic field, we can use Ampere's law to relate the magnetic field to the current. Ampere's law states that the magnetic field around a closed loop is directly proportional to the current passing through the loop.
The magnetic field created by the loop at point (0, y, 0) is given by:
B_loop = µ₀⋅I_loop / (2⋅π⋅r)
Where:
Since the loop is very small, we can consider a small section of it with length dy. The current through this section is given by dI = I_loop / (2⋅π⋅r), and the magnetic field created by this section is given by dB = µ₀⋅dI / (2⋅π⋅r).
The net magnetic field at point (0, y, 0) is the sum of the magnetic fields created by all small sections of the loop. Integrating both sides of the equation, we get:
∫(B_loop)dy = ∫(µ₀⋅dI / (2⋅π⋅r))dy
B⋅y = µ₀⋅(I_loop / (2⋅π⋅r))⋅y
B = µ₀⋅(I_loop / (2⋅π⋅r))
Comparing this equation to B = k⋅y, we can determine the constant k:
k = µ₀⋅(I_loop / (2⋅π⋅r))
Substituting the given values: k = (4π × 10^-7 T⋅m/A)⋅(4 A / (2⋅π⋅0.1 m))
Calculating: k = 0.04 T/A
Therefore, the constant k is 0.04 T/A.
b) To calculate the magnetic field at point P(0, 0.1, 0) m, we can use the equation B = k⋅y and substitute the value of y:
B = 0.04 T/A ⋅ 0.1 m = 0.004 T
Therefore, the magnetic field at point P is 0.004 T.
c) To calculate the net magnetic force experienced by the small loop, we can use the equation:
F = ∫(I_loop⋅dL ⨯ B_wire)
Where:
Since the small loop lies in the y-z plane, the force on each small segment dL of the small loop will be perpendicular to both dL and B_wire. The net force will be in the x-direction.
Let's consider a small segment of the wire loop at a position (0, y, 0) m. The force on this segment is given by:
dF = I_loop⋅(dL)⋅(B_wire)⋅sin(θ)
Where θ is the angle between dL and B_wire.
Since dL and B_wire are in the y-direction and magnetic field B_wire = k⋅y, the angle θ is 90°. Therefore, sin(θ) = sin(90°) = 1.
dF = I_loop⋅(dL)⋅(B_wire)
The magnetic field B_wire = k⋅y, so:
dF = I_loop⋅(dL)⋅(k⋅y)
Integrating both sides, we get:
F = ∫(I_loop⋅(dL)⋅(k⋅y))
The integral on the right side represents the sum of the forces on each small segment of the loop. Substituting the values:
F = ∫(2 A⋅(k⋅y)⋅dL)
Since the loop is symmetric, the forces on opposite segments will cancel each other out. Therefore, we only need to consider half of the loop.
F = 2⋅∫(k⋅y⋅dL)
The integral ∫(y⋅dL) represents the area enclosed by the loop, which is equal to the area of a circle of radius 0.05 m.
F = 2⋅k⋅(π⋅(0.05 m)^2)
Substituting the value of k and calculating:
F = 2⋅0.04 T/A ⋅ (π⋅(0.05 m)^2)
F ≈ 0.01 N
Therefore, the magnitude of the net magnetic force experienced by the small loop is approximately 0.01 N, and the force is directed along the positive x-axis.