Electromagnetism - Conductor

Two long conductors are arranged as shown above to form overlapping cylinders, each of radius r, whose centers are separated by a distance d. Current of density J flows into the plane of the page along the shaded part of one conductor and equal current flows out of the plane of the page along the shaded portion of the other, as shown. What are the magnitude and direction of the magnetic field at point A?

A. (µ₀ ⁄ 2π)πdJ, in the +y-direction
B. (µ₀ ⁄ 2π)d²J ⁄ r, in the +y-direction
C. (µ₀ ⁄ 2π)4d²J ⁄ r, in the −y-direction
D. (µ₀ ⁄ 2π)J²r ⁄ d, in the −y-direction
E. There is no magnetic field at A

(GR9677 #69)

Solution:

Using right-hand rule, the B direction is in the +y direction
→ C, D, E are FALSE

For option (B), if r → 0,  B → ∞, B cannot be infinite.

Answer: A

Electromagnetism - Lenz's Law

Questions 54-55.

A rectangular loop of wire with dimensions shown above is coplanar with a long wire carrying current I. The distance between the wire and the left side of the loop is r. The loop is pulled to the right as indicated. 

What are the directions of the induced current in the loop and the magnetic forces on the left and the right sides of the loop as the loop is pulled? 

	Induced Current	Force on Left Side	Force on Right Side
A.	Counterclockwise	To the left	To the right
B.	Counterclockwise	To the left	To the left
C.	Counterclockwise	To the right	To the left
D.	Clockwise	To the right	To the left
E.	Clockwise	To the left	To the right

(GR9277 #54)

Solution:

Lenz Law: The tendency of nature to resist any change in magnetic flux passing through a loop of wire.

When the loop is pulled away, it feels a decreasing current and resists the change of magnetic flux by inducing an increasing current in the same direction as I. 

→ The induced current in the loop is clockwise. A, B, C are FALSE.

Magnetic force between two currents

Since the current in the loop is clockwise:

• current in the left side of the loop is parallel to the long wire, they are attracted to each other → the force on the left is to the left.
• current in the right side of the loop is antiparallel to the long wire, they are repelled to each other → the force  on the left is to the right.

Answer: E

Electromagnetism - Lorentz Force

A negative test charge is moving near a long straight wire in which there is a current. A force will act on the test charge in a direction parallel to the direction of the current if the motion of the charge is in a direction

A. Toward the wire
B. Away from the wire
C. The same as that of the current
D. Opposite to that of the current
E. Perpendicular to both the direction of the current and the direction toward the wire

(GR8677 #29)

Solution:

Right Hand Rule #1

 

In this case, the force is parallel to the current,

 

Lorentz Force:

Right hand rule #2:

 

→ For positive charge:

→ For negative charge:

 

Therefore, the motion of the charge is in a direction toward the wire:

 

Answer: A



Alternative Solution:

 

Let the current points upwards along the page →
From the left side of the current, magnetic field is coming out of the page → 
From the right side of the current, magnetic field is going into the page → 

Lorentz Force:
Negative test charge →
If force parallel to current →
Magnetic field (on the right side) → 
Direction of ?



Using the same method, from left side: