Physics 1502: Lecture 17Today’s Agenda

• Announcements:– Midterm 1 distributed today

• Homework 05 due FridayHomework 05 due Friday

• Magnetism

Trajectory in Constant B Field

FFv

R

x x x x x x

x x x x x x

x x x x x xv B

q

x x x x x x

x x x x x x

x x x x x x

• Suppose charge q enters B field with velocity v as shown below. (vB) What will be the path q follows?

• Force is always to velocity and B. What is path? – Path will be circle. F will be the centripetal force needed to

keep the charge in its circular orbit. Calculate R:

Radius of Circular Orbit

• Lorentz force:

• centripetal acc:

• Newton's 2nd Law:

x x x x x x

x x x x x x

x x x x x xv

F

B

qFv

R

x x x x x x

x x x x x x

x x x x x x

This is an important result, with useful experimental consequences !

Ratio of charge to mass for an electron e-

3) Calculate B … next week; for now consider it a measurement

4) Rearrange in terms of measured values, V, R and B

1) Turn on electron ‘gun’

V

‘gun’

2) Turn on magnetic field B

R

&

Lawrence's Insight"R cancels R"

• We just derived the radius of curvature of the trajectory of a charged particle in a constant magnetic field.

• E.O. Lawrence realized in 1929 an important feature of this equation which became the basis for his invention of the cyclotron.

• R does indeed cancel R in above eqn. So What??– The angular velocity is independent of R!!

– Therefore the time for one revolution is independent of the particle's energy!

– We can write for the period, T=2/ or T = 2m/qB– This is the basis for building a cyclotron.

• Rewrite in terms of angular velocity !

The Hall Effect

cd

l

a

c

B

BI

I-

vd F

Hall voltage generatedacross the conductor

qEH

Force balance

Hd qEBqv =

Hd EBv =

BdvdEV dHH ==

Using the relation between drift velocity and current we can write:

tcoefficien Hall- /1 , nqRl

IBR

nql

IB

nqA

IBdBdvV H

HdH =====

The Laws of Biot-Savart & Ampere

x

Rr

P

Idx

dl

I

Calculation of Electric Field

• Two ways to calculate the Electric Field:

• Coulomb's Law:

• Gauss' Law

• What are the analogous equations for the Magnetic Field?

"Brute force"

"High symmetry"

Calculation of Magnetic Field

• Two ways to calculate the Magnetic Field:

• Biot-Savart Law:

• Ampere's Law

• These are the analogous equations for the Magnetic Field!

"Brute force" I

"High symmetry"

Biot-Savart Law…bits and pieces

I

dl

dBX

r

So, the magnetic field “circulates” around the wire

B in units of Tesla (T)

A

0= 4X 10-7 T m /A

Magnetic Field of Straight Wire

• Calculate field at point P using Biot-Savart Law:

• Rewrite in terms of R,:

x

Rr

P

IdxWhich way is B?

Magnetic Field of Straight Wire

x

Rr

P

Idx

1

Lecture 17, ACT 1• I have two wires, labeled 1 and 2, carrying equal

current, into the page. We know that wire 1 produces a magnetic field, and that wire 2 has moving charges. What is the force on wire 2 from wire 1 ?

(a) Force to the right (b) Force to the left (c) Force = 0

Wire 1

IX

Wire 2

IX

B

F

Force between two conductors

• Force on wire 2 due to B at wire 1:

• Total force between wires 1 and 2:

• Force on wire 2 due to B at wire 1:

• Direction:attractive for I1, I2 same direction

repulsive for I1, I2 opposite direction

Circular Loop

x

•

z

R

R

• Circular loop of radius R carries current i. Calculate B along the axis of the loop:

• Magnitude of dB from element dl:

r dB

r

z

dB

• What is the direction of the field?• Symmetry B in z-direction.

Circular Loop

• Note the form the field takes for z>>R:

• Expressed in terms of the magnetic moment:

note the typical dipole field behavior!

x

•

z

R

R

r

r

dB

dB

z

Circular Loop

0 3

x =

y =

R

B

z

z

00

1

z3

Lecture 17, ACT 2• Equal currents I flow in identical

circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction.– What is the magnetic field Bz(A)

at point A, the midpoint between the two loops?

(a) Bz(A) < 0 (b) Bz(A) = 0 (c) Bz(A) > 0

x

o x

o

z

I I

A B

Lecture 17, ACT 3• Equal currents I flow in identical

circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction.

(a) Bz(B) < 0 (b) Bz(B) = 0 (c) Bz(B) > 0

– What is the magnetic field Bz(B) at point B, just to the right of the right loop?

x

o x

o

z

I I

A B

Magnetic Field of Straight Wire• Calculate field at distance R

from wire using Ampere's Law:

• Ampere's Law simplifies the calculation thanks to symmetry of the current! ( axial/cylindrical )

dl RI

• Choose loop to be circle of radius R

centered on the wire in a plane to wire. – Why?

» Magnitude of B is constant (fct of R only)» Direction of B is parallel to the path.

– Current enclosed by path = I

– Evaluate line integral in Ampere’s Law:

– Apply Ampere’s Law:

• What is the B field at a distance R, with R<a (a: radius of wire)?

• Choose loop to be circle of radius R,

whose edges are inside the wire.

– Current enclosed by path = J x Area of Loop

B Field inside a Long Wire ?

RI

Radius a

– Why?» Left Hand Side is same as before.

– Apply Ampere’s Law:

Review: B Field of aLong Wire

B = 0 I

2 ra2

• Inside the wire: (r < a)

• Outside the wire: (r>a)

B = 0 I

2 r

0 4

x =

y =

r

B

a

Lecture 17, ACT 4• A current I flows in an infinite straight

wire in the +z direction as shown. A concentric infinite cylinder of radius R carries current I in the -z direction. – What is the magnetic field Bx(a) at

point a, just outside the cylinder as shown?

2A

(a) Bx(a) < 0 (b) Bx(a) = 0 (c) Bx(a) > 0

Lecture 17, ACT 4• A current I flows in an infinite straight

wire in the +z direction as shown. A concentric infinite cylinder of radius R carries current I in the -z direction.

2B

(a) Bx(b) < 0 (b) Bx(b) = 0 (c) Bx(b) > 0

– What is the magnetic field Bx(b) at point b, just inside the cylinder as shown?

B Field of a Solenoid

• A constant magnetic field can (in principle) be produced by an sheet of current. In practice, however, a constant magnetic field is often produced by a solenoid.

• If a << L, the B field is to first order contained within the solenoid, in the axial direction, and of constant magnitude. In this limit, we can calculate the field using Ampere's Law.

L• A solenoid is defined by a current I flowing

through a wire which is wrapped n turns per unit length on a cylinder of radius a and length L.

a

B Field of a Solenoid

• To calculate the B field of the solenoid using Ampere's Law, we need to justify the claim that the B field is 0 outside the solenoid.

• To do this, view the solenoid from the

side as 2 current sheets.

xxx xx

•• • ••• The fields are in the same direction in the

region between the sheets (inside the solenoid) and cancel outside the sheets (outside the solenoid).

xxx xx

•• • ••• Draw square path of side w:

(n: number ofturns per unitlength)

Toroid• Toroid defined by N total turns

with current i.

• B=0 outside toroid! (Consider integrating B on circle outside toroid)

• To find B inside, consider circle of radius r, centered at the center of the toroid.

x

x

x

x

x

x

x

x

x x

x

x x

x

x

x

•

•

•

•

• •

•

••

•

• •

•

•

••

r

B

Apply Ampere’s Law:

Magnetic FluxDefine the flux of the magnetic field through a surface (closed or open) from:

Gauss’s Law in Magnetism

dS

B B

Magnetism in Matter• When a substance is placed in an external magnetic field Bo,

the total magnetic field B is a combination of Bo and field due to magnetic moments (Magnetization; M):

– B = Bo + oM = o (H +M) = o (H + H) = o (1+) H

» where H is magnetic field strength is magnetic susceptibility

• Alternatively, total magnetic field B can be expressed as:– B = m H

» where m is magnetic permeability» m = o (1 + )

• All the matter can be classified in terms of their response to applied magnetic field:

– Paramagnets m > o

– Diamagnets m < o

– Ferromagnets m >>> o