1 MAGNETISM

A magnetic field is a vector field that permeates space and which can exert a magnetic force on moving

electric charges and on magnetic dipoles.

We define the magnitude of the magnetic field by measuring the magnetic force on a moving cha rge q:

B = 1 T(Tesla) =

Direction at any location is the direction in which the north pole of the compass needl e points at that location. Outside magnet: N → S Inside magnet: S → N (always closed loops)

1. An electric charge experiences a magnetic force when moving in a magnetic field. Magnetic force acting on a charge q Magnetic force on a wire carrying current I

in a magnetic Field B: F = qvB sinin a magnetic field B: F = I LB sin q = charge [C] I = current [A] v = velocity [m/s] L = length [m] B = magnetic field [Tesla T] B = magnetic field [T] = angle between v and B = angle between I and B

R-H-R 1: The direction of the magnetic force on a charge/current is given by the right-hand rule 1:

Outstretch fingers in the direction of v (or current I). Curl fingers as if rotating vector v (I ) into vector B.

Magnetic force on a positive charge (or I) is in

the direction of the thumb. Magnetic force on a negative charge points in opposite direction.

Charge q in elec. field E and mag. field B

The electric force: Felec = Eq

● is always parallel to the direction of the electric field.

● acts on a charged particle independent of the particle’s velocity (even at rest).

● does the work when moving charge.

The work, W = Fel d cos θ1, is converted into kinetic energy which is, in the case of conductors, transferred to thermal energy through collisions with the lattice ions causing increased amplitude of vibrations seen as temperature rise.

The magnetic force: Fmag = qvB sin

● is always perpendicular to the direction of the magnetic field

● acts on a charged particle only when the particle is in motion and only if v and B do not point in the same or opposite direction (sin 00 = sin 1800 = 0).

● Force is perpendicular to the direction of the motion, so the work done by magnetic force is zero.

W = Fmag d cos θ1 = 0 (cos 900 = 0).

W = ΔKE = 0

Hence change in kinetic energy of the charge is 0, and that means that mag. force cannot change the speed of the charge. Magnetic force can only

change direction of the velocity – therefore it acts as centripetal force.

θ1 is angle between F and direction of motion

2 Examples of the Lorentz Force

Two important applications of the Lorentz force are 1) the trajectory of a charged particle in a uniform magnetic field and 2) the force on a current-carrying conductor.

1) The trajectory of a charge q in a uniform magnetic field B

● Force is perpendicular to B,v ● Magnetic force does no work! (W = F d cos θ1 = 0 ) ● Speed is constant (W = Δ KE = 0 ) ● Circular motion Centripetal force definition: Motion in a circle represents accelerated motion, and requires a force directed toward the center of the circle. This force is called the centripetal force which means "center seeking" force. It is forcing mass m to move in the circle of radius R with the speed v . Centripetal force force has the magnitude

Charged particle in a magnetic field when v B: In the case the charge q is subject to the uniform field B, centripetal force Fc is magnetic force forcing the charge to move in a circle:

B = magnetic field [T] is represented by the crosses – into the page

R =is the radius of the path F is magnetic force on the charge directed toward the centre of the circular path m = mass [kg] v = velocity [m/s] q = charge [C]

● massive or fast charges – large circles ● large charges and/or large B – small circles

Positive charge q in magnetic field B

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APPLICATION: A mass spectrometer – the path shown is for positive ions

Step 0 – if we want to find a mass of molecules

or atoms we have to ionize them first. Step 1 – Acceleration through potential difference ΔV

ΔU = q ΔV = ½ mv2

v =

tep 2 - the velocity selector – Crossed fields

The ions emerge from the acceleration stage with a range of speeds (different masses – different accelerations). Velocity selector/crossed fields is designed to allow ions of only a particular velocity to pass through undeflected. Two forces are acting on the charge q

Felec = qE and Fmag = qvB

if qvB = qE the charge moves through crossed fields undeflected

so only ions with speed v = E/B are undeflected and will move in the straight l ine

Step 3 - mass separation

All these ions, with the same charge and velocity, enter the mass separation stage - a region with a uniform magnetic field. Magnetic field will force the ion entering it with velocity perpendicular to it, to travel in a circular path. The only thing different for these particles

is the mass, so the heavier ions travel in a circular path of larger radius than the lighter ones.

Charged particle in a magnetic field when v B:

R

mvqvB

2

qB

mvR

2. A moving charge produces a magnetic field.

R-H-R 2: The direction of the magnetic field produced by electric current is given by the right-hand rule 2: If a wire is grasped in the right hand with the thumb in the direction of current flow, the fingers will curl in the direction of the magnetic field.

massive or fast charges – large circles large charges and/or large B – small circles

4 Magnetic field B around a wire with current I Magnetic Field B Inside of a Solenoid

B = B = 0 n I

= the permeability of free space 4×10-7 T·m/A I = current [A] r = distance from the center of the conductor n = N/L number of turns of wire per unit length

Magnetic Force per unit length between Two Parallel Wires: d

II

L

F

L

F 21021

2

Current I1 creates magnetic field B1 Place a wire with current I2 at distance d from the wire at distance d from I1

B1 =

(a) two parallel currents (b) two antiparallel curren ts

– the force between them is attractive – the force between them is repulsive

One Ampere is defined as that current flowing in each of two infinitely-long parallel wires of negligible cross-sectional area separated by a distance of one metre in a vacuum that results in a force of exactly 2 x 10-7 N per metre of length of each wire.

Electricity can be produced from many different types of energy, and all of these methods make

use of a generator to convert mechanical energy to electrical energy. Electrical generator

Electrical energy is usually produced by rotating coil in a uniform magnetic field. Free electrons in a wire will experience magnetic force if the wire moves in a magnetic field; the electrons inside it experience a force causing them to move to one end of the wire. This causes a potential difference along the wire which can be used to create current in a circuit. We say that current has been induced in the wire.

The force on I2 is:

F2 = L I2 B1 = L I2

F2 = L

The magnetic field is concentrated into a nearly uniform field in the centre of a long solenoid. The field outside is weak and diverging

5 A generator uses the same principle, but instead of the wire moving in a straight line, a coil rotates in the magnetic field. As the coil rotates, resulting in a current that keeps changing direction, called an alternating current or AC for short. To prevent the wires from getting twisted when the coil rotates, there must be sliding contacts between the coil and circuit. However, friction between these sliding components will eventually wear away the contacts. An alternative arrangement is for the magnets to rotate inside the coils, this also results in an alternating current but has no sliding parts

Electric Motor You find it everywhere: turntables, washing machined, any fan, air conditioners, wherever something is moving there is an electric motor. There are many coils of wires, but b/c of simplicity we drew only one. Each end of the coil is attached to a metallic half-ring. Rubbing against each of the half-rings is a graphite contact called a brush. Half–rings rotate with the coil; the graphite brushes remain stationary. Left picture: the current from the battery enters the coil through the left brush, goes around the coil, and leaves through the right half-ring and brush. Forces on two vertical sides result in rotation (turning) of the coil. The coil rotates until it riches the position shown in the middle picture. In this position t he half rings momentarily lose electrical contact with the brushes: no current – no force, no ability of the force to cause rotation (so called torque). But, b/c of its rotational inertia (moving object does not stop momentarily) the coil continues to rotate, and half-rings reestablish contact with the brushes. So there is again current in the coil, but now in opposite direction (the side that had current upwards, now has current downwards b/c it is connected to different terminal). The split ring ensures that the current is always in the proper direction such that the torque produces a continuous rotation of the coil.

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PRACTICE MAGNETISM 1

1. Where is the magnitude of the magnetic field around a permanent magnet greatest? a . close to the poles

b. far from the poles c. The magnitude is equal at all points on the field. d. The magnitude depends on the material of the magnet.

2. Which compass needle orientation in the figure might

correctly describe the magnet’s field at that point? a . a b. b c. c d. d

3. If a proton is released at the equator and falls toward Earth under the influence of gravity, the magnetic force on the proton

wi l l be toward the a . north. b. south.

c. east. d. west.

4. What i s the path of an electron moving perpendicular to a uniform magnetic field?

a . a s tra ight line b. a ci rcle c. an ellipse d. a parabola

5. What i s the path of an electron moving parallel to a uniform magnetic field? a . s tra ight l ine b. ci rcle c. el lipse d. parabola

6. A proton moves past a bar magnet as shown. Find the direction of the force it experiences in each case.

7. An electron is moving with a speed of 3.0 x105 ms -1 in a direction that is at right angles to a uniform magnetic field of 3.0 x 10-3 T. Ca lculate

a . the force exerted on the electron. b. the radius of the path of the electron.

9. The magnetic field of a bar magnet is shown in the figure. Is the magnet’s north pole at A or B?

10. Find the direction of the force on an proton moving through the magnetic field shown .

11.Find the direction of the force on an proton moving through the magnetic field shown .

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12. Find the direction of the force on an electron moving through the magnetic field shown .

13. Find the direction of the force on an electron moving through the magnetic field shown .

14. Find the direction of the force on an electron moving through the magnetic field shown .

15. A negative charge is moving through a magnetic field. The direction of motion and the direction of the force acting on i t at one moment are shown in the figure. Find the direction of the magnetic field.

16. A negative charge is moving through a magnetic field. The direction of motion and the direction of the force acting on i t at one moment are shown in the figure. Find the direction of the magnetic field.

17. A negative charge is moving through a magnetic field. The direction of motion and the direction of the force acting on it at one moment are shown in the figure. Find the direction of the magnetic field.

18. An electron moves north at a velocity of 9.8 104 m/s and has a magnetic force of 5.6 10–18 N exerted on i t. If the magnetic

field points upward, what is the magnitude of the magnetic field?

19. An electron moves north at a velocity of 2.7 104 m/s and has a magnetic force of 9.5 10–18 N exerted on i t. If the magnetic

field points upward, what is the magnitude of the magnetic field?

20. A charged particle i s injected into a region of uniform magnetic field and travels in a circular arc.

If the particle were to be injected with a greater speed, what would be true of the magnetic force on i t and the radius of its path?

Force Arc radius

A. greater greater B. greater smaller C. smaller greater D. smaller smaller

8 12. The diagram below shows a charged particle about to enter a region of uniform magnetic field directed into the page.

Which of the following correctly describes the change, if any, in the kinetic energy and the momentum of the particle in the magnetic field? Kinetic energy Momentum A. Changed Changed B. Changed Unchanged

C. Unchanged Changed D. Unchanged Unchanged

22. A charged particle of mass m and charge q i s travelling in a uniform magnetic field with speed v such that the magnetic force on

the particle is F. The magnetic force on a particle of mass 2m, charge q and speed 2v travelling in the same direction in the magnetic field is A. 4F. B. 2F. C. F.

D. ½ F.

23. The diagram show a straight wire carrying current i in a uniform magnetic field. The magnetic force on the wire is indicated by

an arrow but the magnetic field is not shown. Of the following possibilities, the direction of the magnet ic field is:

24. What i s the direction of a magnetic field in each of the four cases that results in a force on the current as shown?

25. A wire that is carrying a current of 3.50 A east has 2.00 m of i ts length in a uniform magnetic f ield of magnetic flux density of

5.00 x 10-7 T di rected vertically into the paper. Determine the magnitude and direction of the force it experiences.

MAGNETISM PRACTICE II

1. A bar magnet is placed in a uniform magnetic field as shown.

(a) Is there a net force on the bar magnet? (b) Will it move? If so, how?

9 2. Find the direction of the missing quantity from B, v and F in each of the cases shown. The circle represents a

positive charge.

3. ( challange) Figure shows two parallel plates with a potential difference of 120 V a distance 5.0 cm apart. The top plate is at the higher potential and the shaded region is a region of magnetic field norma l to the page. (a) What should the magnetic field magnitude and direction be such that an electron experiences zero net force

when shot through the plates with a speed of 2 x 105 ms -1 (b) Would a proton shot with the same speed through the plates experience zero net force? (c) If the electron’s speed were doubled, would it sti l l be undeflected if the magnetic field took the value you found

in (a)?

4. A wire that is carrying a current of 3.50 A east has 2.00 m of its length in a uniform magnetic field of magnetic flux density of 5.00 x 10-7 T directed vertically into the paper.

Determine the magnitude and direction of the force it experiences.

5. An electron is moving with a speed of 3.0 x105 ms -1 in a direction that is at right angles to a uniform magnetic field

of 3.0 x 10-3 T. Calculate a. the force exerted on the electron. b. the radius of the path of the electron.

8. An electron is shot along the axis of a solenoid that carries current. Will it experience a magnetic force?

9. What i s the direction of the magnetic field at points P and Q in the plane of a ci rcular loop

carrying a counterclockwise current, as shown?

10. A suitable unit of magnetic field strength is A. A N-1 m-1

B. kg s -2 A-1 C. A m N-1 D. kg A s2

11. The diagram below shows a point P on the Earth’s surface at which a compass needle is suspended freely.

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Which one of the following gives the correct direction in which the needle of the compass will point?

14. (challange) A proton is in a region where a uniform electric field of 5 × 104 V/m is perpendicular to a uniform magnetic field of 0.8 T. If i ts acceleration is zero then i ts speed must be:

A) 0 B) 6.3 × 104 m/s

C) 1.6 × 104 m/s D) 4.0 × 105 m/s

E) any va lue but 0

15. A long wire that carries a current I i s bent into five loops as shown in the figure.

If the observer could "see" the magnetic field inside this arrangement of loops, how would i t appear?

16. The currents in two parallel wires are I and 3I in the directions shown in the diagram below.

The magnetic force on wire 2 due to the current in wire 1 is F. The magnitude of the force on wire 1 due to the current in wire 2 is

A. F/3. B. F/2. C. F. D. 3F.

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17. Current in a loop is counter clock wise. In which direction is induced magnetic field inside the loop and outside the loop in the plane of paper.

18. Electrons are going around a circle in a counterclockwise direction as shown. At the center of the circle they produce a magnetic field that is:

A) into the page

B) out of the page C) to the left D) to the right

E) zero

19.A long straight wire carries current as shown. Two electrons move with velocities that are parallel and perpendicular

to the current. Find the direction of the magnetic force experienced by each electron.

21. The diagram below shows three parallel wires P, Q and R that are

equally spaced.

The currents in the wires are each of the same magnitude I and are in the directions shown. The resultant force on wire Q due to the current in wire P and in wire R is

A. perpendicular and into the plane of the paper. B. perpendicular and out of the plane of the paper. C. in the plane of the paper to the right. D. in the plane of the paper to the left.

22. A long, straight current-carrying wire is placed normal to the plane of the page. The current in the wire is into the

plane of the page.

12 Which of the following diagrams best represents the magnetic field around the wire?

23. Two long parallel straight wires carry equal currents in opposite directions. At a point midway between the wires, the magnetic field they produce is:

A) zero B) non-zero and along a l ine connecting the wires

C) non-zero and parallel to the wires D) non-zero and perpendicular to the plane of the two wires E) none of the above

24. Magnetic field l ines inside the solenoid shown are: A) clockwise circles as one looks down the axis from the top of the page

B) counterclockwise circles as one looks down the axis from the top of the page C) toward the top of the page D) toward the bottom of the page E) in no direction since B = 0

19. A wire, connected to a battery and switch, passes through the center of a long current-carrying solenoid as shown in the drawing.

When the switch is closed and there is a current in the wire, what happens to the portion of the wire that runs inside

of the solenoid? A. There is no effect on the wire. B. The wire is pushed downward. C. The wire is pushed upward.

D. The wire is pushed into the plane of the paper. E. The wire is pushed out of the plane of the paper.