Question

A loop of wire in the shape of a rectangle of width w and length L and a long, straight wire carrying a current I lie on a tabletop as shown in the figure below.
A long, straight, horizontal wire carries current I toward the right. A rectangular loop of wire, with length L and height w, lies below the straight wire. The length is parallel to the straight wire, and the top edge of the loop is a distance h below the straight wire.
(a)
Determine the magnetic flux through the loop due to the current I. (Use any variable stated above along with the following as necessary: μ0.)
ΦB =
(b)
Suppose the current is changing with time according to I = a + bt, where a and b are constants. Determine the magnitude of the emf (in V) that is induced in the loop if b = 14.0 A/s, h = 1.00 cm, w = 20.0 cm, and L = 1.05 m.
V
(c)
What is the direction of the induced current in the rectangle?
clockwise counterclockwise The magnitude is zero.
What If? Suppose a constant current of I = 6.00 A flows in the straight wire and the loop moves from an initial position
h0 = 1.00 cm
toward the bottom of the figure at a constant speed of v = 16.0 cm/s.
(e)
What is the direction of the induced current in the loop 1.00 s after it begins to move?
clockwise counterclockwise The magnitude is zero.

Answer 1

The magnetic flux through the loop due to the current I following BAcosθ is zero.

The magnetic flux through the loop can be determined using the equation:

Φ = BAcosθ

where Φ is the magnetic flux, B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop.

To determine the magnetic flux through the loop, we need to first calculate the magnetic field strength at the location of the loop. The magnetic field strength at a point due to a straight wire carrying a current I is given by:

B = µ0I/2πr

where µ0 is the permeability of free space, I is the current in the wire, and r is the distance from the wire to the point where the field strength is being calculated.

In this case, the distance from the wire to the top edge of the loop is h, so the magnetic field strength at the top edge of the loop is:

B = µ0I/2πh

The area of the loop is A = wL, and the angle between the magnetic field and the normal to the loop is θ = 90°, since the field is perpendicular to the loop. Therefore, the magnetic flux through the loop is:

Φ = (µ0I/2πh) * wL * Cos(90°)

= (µ0I/2πh) * wL * 0

= 0

Therefore, the magnetic flux through the loop is zero.

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