Magnetism

April 19, 2023

Magnets and Magnetic Fields

Magnets cause space to be modified in their vicinity, forming a “magnetic field”.

The magnetic field caused by magnetic “poles” is analogous to the electric field caused by electric “poles” or “charges”.

Magnets and Magnetic Fields

The north pole is where the magnetic field lines leave the magnet, and the south pole is where they reenter.

Magnetic field lines differ from electric field lines in that they are continuous loops with no beginning or end.

Magnetic Field, B

Magnetic “Monopoles” do not exist

This is another way that magnetic fields differ from electric fields.

Magnetic poles cannot be separated from each other in the same way that electric poles (charges) can be.

Units of Magnetic Field

Tesla– N/(C m/s)– N/(A m)

Gauss– 1 Tesla = 104 gauss

Magnetic Force on Particles

Magnetic fields cause the existence of magnetic forces.

A magnetic force is exerted on a particle within a magnetic field only if: the particle has a charge and the charged particle is moving with at least a

portion of its velocity perpendicular to the magnetic field.

Magnetic Force on a Charged Particle

q: charge in Coulombsv: speed in meters/secondB: magnetic field in Teslaθ: angle between v and B

direction: Right Hand Rule

The Right Hand Rule

1. Align your hand along the first vector.

2. Orient your wrist so that you can “cross” your hand into the second vector.

3. Your thumb gives you the direction of the third vector (which is the result).

Sample Problem: Calculate the magnitude force exerted on a 3.0 μC charge moving north at 300,000 m/s in a magnetic field of 200 mT if the field is directed

North.

South.

East.

West.

Magnetic forces…

are always orthogonal (at right angles) to the plane established by the velocity and magnetic field vectors.

can accelerate charged particles only by changing their direction.

can cause charged particles to move in circular or helical paths.

Magnetic forces cannot...

…change the speed or kinetic energy of charged particles.

do work on charged particles.

Magnetic Forces…

…are centripetal.

Remember that centripetal acceleration is v2/r.

Centripetal force is therefore mv2/r.

Sample Problem: Draw the free body diagram and draw the path that the charged particle will take as it moves through the B-field. Ignore gravitational effects.

Sample Problem: An electric field of 2000 N/C is directed to the south. A proton is traveling at 300,000 m/s to the west. What is the magnitude and direction of the force on the proton? Describe the path of the proton? Ignore gravitational effects.

Sample Problem: A magnetic field of 2000 mT is directed to the south. A proton is traveling at 300,000 m/s to the west. What is the magnitude and direction of the force on the proton? Describe the path of the proton? Ignore gravitational effects.

Sample Problem: Calculate the force and describe the path of this electron.

Sample problem: Calculate the force and describe the path of this electron.

iClicker

Which of the paths above represents the path of an electron traveling without any loss of energy through a uniform magnetic field directed into the page?

(A) A (B) B (C) C (D) D (E) E

iClicker

A wire in the plane of the page carries a current directed toward the top of the page as shown above. If the wire is located in a uniform magnetic field B directed out of the page, the force on the wire resulting from the magnetic field is

(A) directed into the page (B) directed out of the page (C) directed to the right (D) directed to the left (E) zero

Sample Problem: How would you arrange a magnetic field and an electric field so that a charged particle of velocity v would pass straight through without deflection?

Sample Problem: It is found that protons traveling at 20,000 m/s pass undeflected through the velocity filter below. What is the magnitude and direction of the magnetic field between the plates?

Magnetic Force on Current -Carrying Wire

F = I L B sin θ I: current in Amperes L: length in meters B: magnetic field in Tesla q: angle between current and field

Sample Problem: What is the force on a 100 m long wire bearing a 30 A current flowing north if the wire is in a downward-directed magnetic field of 400 mT?

Sample Problem: What is the magnetic field strength if the current in the wire is 15 A and the force is downward and has a magnitude of 40 N/m? What is the direction of the current?

Magnetic Fields…

Affect moving charge F = q v B sin θ F = I L B sin θ

Hand rule is used to determine direction of this force.

They are also caused by moving charge – Lenz’s Law

Right Hand Rule for Straight Currents:

1. Curve your fingers

2. Place your thumb in direction of current.

3. Curved fingers represent curve of magnetic field.

4. Field vector at any point is tangent to field line.

iClicker

The direction of the magnetic field at point R caused by the current I in the wire shown above is

(A) to the left (B) to the right (C) toward the wire (D) into the page (E) out of the page

Magnetic Field produced by straightcurrents.

Magnitude of Magnetic Field for straight currents.

μo: 4π × 10-7 T m / A magnetic permeability of free space

I: current (A) r: radial distance from center of wire (m)

r

IB o

2

Sample Problem: What is the magnitude and direction of the magnetic field at point P, which is 3.0 m away from a wire bearing a 13.0 A current?

Principle of Superposition

When there are two or more currents forming a magnetic field, calculate B due to each current separately and then add them together using vector addition.

iClicker

Two long, parallel wires, fixed in space, carry currents I1 and I2. The force of attraction has magnitude F. What currents will give an attractive force of magnitude 4F?

(A) 2 I1 and ½ I2 (B) I1 and ¼ I2 (C) ½ I1 and ½ I2 (D) 2 I1 and 2 I2 (E) 4 I1 and 4 I2

Sample Problem: What is the magnitude and direction of the electric field at point P if there are two wires producing a magnetic field at this point?

Sample Problem: Where would the magnetic field be zero?

Solenoid

A solenoid is a coil of wire.

When current runs through the wire, it causes the coil to become an “electromagnet”.

Air-core solenoids have nothing inside of them.

Iron-core solenoids are filled with iron to intensify the magnetic field.

Right Hand Rule for Curved Wires

1. Curve your fingers.

2. Place them along wire loop so that your fingers point in direction of current.

3. Your thumb gives the direction of the magnetic field in the center of the loop, where it is straight.

4. Field lines curve around and make complete loops.

Sample Problem: What is the direction of the magnetic field produced by the current I at A? At B?

Magnetic Flux

The product of magnetic field and area.

Can be thought of as a total magnetic “effect” on a coil of wire of a given area.

Maximum Flux

The area is aligned so that a perpendicular to the area points parallel to the field

Magnetic Flux

ΦB = B A cos θ ΦB: magnetic flux in Webers (Tesla

meters squared) B: magnetic field in Tesla A: area in meters squared. θ: the angle between the area and the

magnetic field.

Sample Problem: Calculate the magnetic flux through a rectangular wire frame 3.0 m long and 2.0 m wide if the magnetic field through the frame is 4.2 mT.a) Assume that the magnetic field is perpendicular to the area vector.b) Assume that the magnetic field is parallel to the area vector.c) Assume that the angle between the magnetic field and the area vector is 30-degrees.

Sample Problem: Assume the angle is 40o, the magnetic field is 50 mT, and the flux is 250 mWb. What is the radius of the loop?

Induced Electric Potential

A system will respond so as to oppose changes in magnetic flux.

A change in magnetic flux will be partially offset by an induced magnetic field whenever possible.

Changing the magnetic flux through a wire loop causes current to flow in the loop.

This is because changing magnetic flux induces an electric potential.

Faraday’s Law of Induction

– ε: induced potential (V)– N: # loops– ΦB: magnetic flux (Webers, Wb)

– t: time (s)

To generate voltage, change B, change A, or change Φ

Sample Problem: A coil of radius 0.5 m consisting of 1000 loops is placed in a 500 mT magnetic field such that the flux is maximum. The field then drops to zero in 10 ms. What is the induced potential in the coil?

iClicker

A square loop of copper wire is initially placed perpendicular to the lines of a constant magnetic field of 5 x 10‑3 tesla. The area enclosed by the loop is 0.2 square meter. The loop is then turned through an angle of 90° so that the plane of the loop is parallel to the field lines. The turn takes 0.1 second. The average emf induced in the loop during the turn is

(A) 1.0 x 10‑4 V (B) 2.5 x 10‑3 V (C) 0.01 V (D) 100 V (E) 400 V

Sample Problem: A single coil of radius 0.25 m is in a 100 mT magnetic field such that the flux is maximum. At time t = 1.0 seconds, field increases at a uniform rate so that at 11 seconds, it has a value of 600 mT. At time t = 11 seconds, the field stops increasing. What is the induced potential

A) at t = 0.5 seconds?B) at t = 3.0 seconds?C) at t = 12 seconds?

Lenz’s Law

The current will flow in a direction so as to oppose the change in flux.

Use in combination with hand rule to predict current direction.

iClicker

In each of the following situations, a bar magnet is aligned along the axis of a conducting loop. The magnet and the loop move with the indicated velocities. In which situation will the bar magnet NOT induce a current in the conducting loop?

Sample Problem: The magnetic field is increasing at a rate of 4.0 mT/s. What is the direction of the current in the wire loop?

iClicker

A magnetic field B that is decreasing with time is directed out of the page and passes through a loop of wire in the plane of the page, as shown above. Which of the following is true of the induced current in the wire loop?

(A) It is counterclockwise in direction. (B) It is clockwise in direction. (C) It is directed into the page.(D) It is directed out of the page. (E) It is zero in magnitude.

Sample Problem: The magnetic field is decreasing at a rate of 4.0 mT/s. The radius of the loop is 3.0 m, and the resistance is 4 W. What is the magnitude and direction of the current?

iClicker

There is a counterclockwise current I in a circular loop of wire situated in an external magnetic field directed out of the page as shown above. The effect of the forces that act on this current is to make the loop

(A) expand in size (B) contract in size (C) rotate about an axis perpendicular to the page(D) rotate about an axis in the plane of the page (E) accelerate into the page

Motional emf

ε = BLv– B: magnetic field (T)– L: length of bar moving through field– v: speed of bar moving through field.

Bar must be “cutting through” field lines. It cannot be moving parallel to the field.

iClicker

A wire of constant length is moving in a constant magnetic field, as shown above. The wire and the velocity vector are perpendicular to each other and are both perpendicular to the field. Which of the following graphs best represents the potential differ ence E between the ends of the wire as a function of velocity?

Sample Problem: How much current flows through the resistor, and in what direction? How much power is dissipated by the resistor?

Sample Problem: How much force is required to keep the rod moving at the constant speed shown?

iClicker

The figure above shows a rectangular loop of wire of width l and resistance R. One end of the loop is in a uniform magnetic field of strength B at right angles to the plane of the loop. The loop is pulled to the right at a constant speed v. What are the magnitude and direction of the induced current in the loop?

Magnitude Direction(A) BlvR Clockwise(B) BlvR Counterclockwise(C) Blv/R Clockwise (D)Blv/R Counterclockwise(E) 0 Undefined