Chapter 28 - the Magnetic Field

Chapter 27 - Magnetic Field and Magnetic Forces

I. Electricity and magnetism essentially developed as separate science until 1820 when Hans Christian Oersted demonstrated that a current produced a magnetic effect. Within 45 years the quantitative connection between electricity and magnetism was demonstrated with the culmination being Maxwell's equations in 1865.

II. Look at a common place where magnetic effects are noticed - permanent magnets

A. Types: bar, horseshoe, cow, u-shaped, alnico, rare earth, etc.

B. Poles of the magnet - preferred regions of attraction and repulsion

1. N = north seeking pole; S = south seeking pole

2. Forces of attraction and repulsion:

"Like poles repel, unlike poles attract."

3. No monopoles.

III. The Magnetic Field

A. A permanent magnet (or a wire carrying an electrical current, as we will see later) alters the space surrounding it. This alteration of the space is called a magnetic field. (Recall the electric field.)

B. Lines of force or lines of induction (just like the lines of force of the electric field) can be drawn to allow us to get a mental picture of the magnetic field.

1. Direction of the field - determined by using a small compass where its north end indicates the direction of the magnetic field.

2. Lines of force – path the compass takes as it is moved in the direction in which the north end points.

3. Strength of the field - indicated by the density of the lines.

27-1 C. Examples of some field lines.

1. North and south poles of one magnet.

2. North and south poles of different magnets.

3. Two north poles.

27-2 27-3 4. Earth's magnetic field.

Declination in SM is 14o E. Dip angle in SM is 56o North Geographic Pole

S NY Chi SM N

North Magnetic Pole: 82.7oN, 114.4o W

 IV. Quantitative definition of the magnetic field strength, :  B  F It is defined similarly to the electric field, E  , except that the charge must be moving. In q examining the motion of a charge in a magnetic field, we find the following:

- the force is proportional to the charge, the velocity and the strength of the magnetic field

- the force on the charge is zero when the charge is moving parallel or antiparallel to the

magnetic field, the maximum force on the charge is when the charge is moving perpendicular

to the field, and the force varies with the sine of the angle between the velocity and the

magnetic field direction.

The definition of the magnetic field must reflect the above observations.

F Define: B  , where  is the angle between the velocity and the magnetic field. qvsin 

27-4 N weber 4 Units: B   Tesla =  10 Gauss A  m m2

27-5 V. Magnetic Flux – like the electric flux, it is a measure of the number of lines of force passing through an area.

       B  B dA  BdAcos  A. If the magnetic field, , is constant, then what is the expression for the magnetic flux? B  A

  B

B. Units of magnetic flux.

C. Examples:  1. A uniform magnetic field, B  0.25iˆ T, is present in space. Find the magnetic flux through each of the surfaces of the triangular block shown.

B 4 m C 3 m

A E x F 2 m 5 m D z

27-6  2. A magnetic field, B  0.50yiˆ T, exists in space. Find the magnetic flux through each of the faces of the cube whose sides are 2 meters long.

3. Qualitative example: Find the flux through a closed area surrounding one end of a bar magnet.

27-7 VI. Force on a charge moving in a magnetic field

A. The definition of the magnetic field can be rewritten as:

   F = qvB sin  or F  qv B

B. Note that the "right hand rule" gives the direction of the force relative to the directions of the velocity and the magnetic field. "The direction of the force is perpendicular to the plane that contains the velocity and the magnetic field."

C. Some directional examples: In each of the following examples, find the direction of the force acting on the charge when a uniform magnetic field points out of the page.  B

1. proton moving to right  v +

_ 2. electron moving down

3. proton moving into page  v +

D. Examples of a charge interacting with a magnetic field.

1. Uniform magnetic field pointing into the page.

v  + B

27-8 2. Velocity selector. Singly ionized atoms are produced in an oven then accelerated through a potential difference. The masses and velocities of the atoms are varied, but the velocity selector selects one particular velocity using crossed electric and magnetic fields.

+ -  + - B + - + - + - + - + - + - + - + -

3. Mass spectrometer - used to identify and separate atoms with small differences in mass.

y  B  -27 5 v 4. A proton (mp = 1.67 x 10 kg, q = +e) has a velocity of 2.00 x 10 m/s at an angle of 30o to the y-axis in the x-y plane. A magnetic 30o field B = 0.400 T pointing in the +x direction is present.

x a. What king of motion does the proton undergo?

b. After one complete revolution, how far is the proton from the origin?

27-9 c. Try this! What is the answer to part b if there is also an electric field present pointing in the +x direction?

27-10 E. Hall effect - an experiment devised by E.H. Hall to find the sign of the current carrier in a conductor. Positive charges are moving in the wire or negative charges are moving in the wire.

d  d B I  B

+ vd A B B A

area, A area, A

Example: A copper strip 2 cm wide and 1 mm thick in a field B = 1.50 T. If the current is 200 A, then what is the Hall potential across the strip? (for copper, n = 8.46 x 1028 atoms/m3)

VI. Magnetic force on a current carrying wire in a magnetic field

A. Since current is charges in motion, and since a force is exerted on moving charges when they are in a magnetic field, then a force should be exerted on a wire carrying a current.  B

B. Straight wire in a uniform magnetic field: find the force I on a straight wire of length L, carrying a current I , when + vd placed in a uniform magnetic field where the wire makes  and angle  with the magnetic field B. Let n = the number of charges per unit volume that are moving with area, A

a velocity vd .

27-11 So, for a straight segment of wire this can be written as:    F  IL  B , where the direction of the force is found by the right hand rule.

C. If the wire is not straight or the magnetic field is not uniform, then forces must be found on short segments of wire and added together. This means integrating the infinitesimal segments. The  differential form of the force on an infinitesimal segment of wire, dl , is given by    dF  Idl  B .

D. Examples:

1. Find the force on a straight 15 cm segment of wire that carries a current of 0.5 A in a uniform magnetic field of 40 G. The segment makes an angle of 30o with the magnetic field.

2. Find the force on a semicircular segment of wire of radius R, carrying a current I, in a uniform magnetic field B. The plane of the semicircle is perpendicular to the magnetic field.

27-12 y

3. A wire lies in the x-y plane, is bent into the shape shown, and placed in a magnetic  field B  kyiˆ, where k is a constant. Find L the force on each straight segment of wire. I x

VIII. Torque on a current loop. A rectangular loop of wire is placed in a uniform magnetic field such that the plane of the loop makes and angle  with the magnetic field.

B B 1 4 1

a right side view front view

A. Find the force on each segment:

27-13 27-14 B. Find the torque on the loop.

C. Total torque.

1. For one turn of wire:

2. For N turns of wire:

3. Define the magnetic moment: magnitude and direction

4. Torque on the loop in terms of the magnetic moment:

D. Examples.

1. A loop consists of ten turns of wire with a radius of 8 cm. A clockwise current of 2 A flows in the wires, and the plane of the loop is in the x-y x plane.

a. Find the magnitude and direction of the magnetic moment.

b. Find the magnitude and direction of the torque on the loop when placed in a magnetic field B = 0.6 T that points in the positive z direction.

c. Find the magnitude and direction of the torque on the loop when placed in a magnetic field B = 0.6 T that points in the negative x direction.

27-15 z

2. A rectangular loop of wire, 30 cm by 40 cm, with a mass B ? per unit length of 10 grams/cm can freely rotate about the x axis. The loop carries a current of 2 A and is place x y in a vertically oriented magnetic field. If the loop makes 30o 40 cm o an angle of 30 with the vertical, then in what is the 30o magnitude and direction of the magnetic field?

3. How does a galvanometer work? Remember that a galvanometer is a current-detecting device.

N S 27-16 4. How does the DC motor work? Main components - poles of magnets, armature, split ring commutator, brushes, DC source.