University

Physics 2

Physics 2

Fall 2024

The following figures show a charge moving through a magnetic field. Sketch the direction of the magnetic force on each.

Use \((\times)\) for a vector into the page, \((\cdot)\) for a vector out of the page, and \(\vec{0}\) for zero force.

\(\vec{F} = q \vec{v} \times \vec{B}\)

- \((\times)\)
- \(\vec{0}\)
- \((\cdot)\)
- \(\vec{0}\)
- \(\leftarrow\)
- \(\nwarrow\)

A square current loop is placed in a uniform magnetic field. Describe the forces on the loop, and its resulting motion.

- The left and right sides of the current feel zero force.
- The top side feels a force out of the page.
- The bottom side feels a force into the page.

The loop will begin to rotate about a horizontal axis through its center, with angular velocity to the right.

When a positive ion flies between the plates of a parallel-plate capacitor, it will be deflected from its original path. Suppose you wanted to prevent the deflection, so the ion travelled straight through. How could you direct a magnetic field to do this?

The electric force on the positive ion would be downward, so the magnetic force would need to be upward. This would require a magnetic field that points into the page \((\times)\) in the region between the plates.

Consider the conducting cube shown, with side length \(a\). If a current flows through the cube while a magnetic field is present, charges may accumulate on the sides of it due to the Hall Effect.

Describe where charges would accumulate in the following cases. (In all cases the magnetic field and the current density are uniform.)

- Positive charge carriers flow in the \(+x\) direction, and \(\vec{B}\) points in the \(+y\) direction.
- Negative charge carriers flow in the \(-z\) direction, and \(\vec{B}\) points in the \(+x\) direction.

\(\vec{F} = q \vec{v} \times \vec{B}\) can be used to determine what direction a charge will be pushed.

- Positive charges on the \(z=a\)
face,

negative charges on the \(z=0\) face. - Positive charges on the \(y=0\)
face,

negative charges on the \(y=a\) face.

Review textbook section 11.6: The Hall Effect.

A uniform magnetic field of magnitude \(B\) is directed parallel to the z-axis. A proton enters the field with a velocity \(\vec{v} = (4 \hat{\jmath} + 3 \hat{k}) \times 10^6\) m/s and travels in a helical path with a radius of \(5.0\) cm.

(You may use a protom mass of \(m_p = 1.672621 \times 10^{-27}\) kg.)

- What is the value of \(B\)?
- What is the time required for one trip around the helix?
- Where is the proton \(5.0 \times 10^{-7}\) s after entering the field?

The helix is a combination of circular motion in the \(xy\) plane (with speed \(4 \times 10^6\) m/s) and constant speed in the \(z\) direction (with speed \(3 \times 10^6\) m/s).

- \(B = 0.835\) T
- \(7.85 \times 10^{-8}\) s
- 40% through the seventh helix loop

Review textbook section 11.3 Motion of a Charged Particle in a Magnetic Field.

The current loop shown in the figure is a parallelogram that lies in the plane of the page. The magnetic field is vertically upward. The loop has a current of \(I=10\) A and the magnitude of the magnetic field is \(B=1.5\) T.

Determine the net force and the net torque on the loop (direction and magnitude).

\(\vec{F} = \vec{0}\) N

\(\vec{\tau} = \vec{\mu} \times \vec{B} = 0.104\) N·m, to the left

The magnetic moment is \(\vec{\mu} = I \vec{A}\).

To magnitude of \(\vec{A}\) is the area of the parallelogram and its direction is in the right-hand sense of the current circulation.

The area of a parallelogram is \(L_1 L_2 \sin \theta\). The direction of \(\vec{A}\) is \(\hat{k}\). (defining the standard \(xyz\) axes).

\(\vec{A} = L_1 L_2 \sin 60^{\circ} \; \hat{k} = (0.00693\) m²\() \; \hat{k}\).

\(\vec{\mu} = (0.0693\) A·m²\() \; \hat{k}\)

Then, with \(\vec{B} = (1.5 \; \rm{T}) \hat{\jmath}\),

\(\vec{\tau} = \vec{\mu} \times \vec{B} = (-0.104\) N·m\() \; \hat{\imath}\).

Review textbook section 11.5 Force and Torque on a Current Loop.

A wire of length \(1.0\) m is wound into a single-turn planar loop. The loop carries a current of \(5.0\) A, and it is placed in a uniform magnetic field of strength \(0.25\) T.

- What is the maximum torque that the loop will experience if it is square?
- If it is circular?
- At what angle relative to \(B\) would the normal to the circular coil have to be oriented so that the torque on it would be the same as the maximum torque on the square coil?

- 0.078 N·m
- 0.10 N·m
- 51.6°

Torque is \(\vec{\tau} = \vec{\mu} \times \vec{B}\), where \(\vec{\mu} = I \vec{A}\). So we need the areas of each loop.

With a given perimeter \(\ell\), you could make a circle with radius given by \(2 \pi r = \ell\), or a square with side length \(\ell/4\).

So the circle area would be \(\displaystyle A_c = \pi r^2 = \pi \left ( \frac{\ell}{2 \pi} \right )^2 = \frac{\ell^2}{4 \pi}\).

The square area would be \(\displaystyle A_s = \left ( \frac{\ell}{4} \right )^2 = \frac{\ell^2}{16}\)

For the angle in part c, solve for \(\theta\) when \(IA_c B \sin \theta = IA_s B\)

(Review section 11.5 Force and Torque on a Current Loop.)

*Last modified: Fri October 18 2024, 04:57 PM.*