Lab #2: Magnetic Fields due to Currents

• Learn how to measure magnetic fields

• Verify biot-savart law/Ampere’s Law by comparing prediction and measurement for various current configurations

• Learn how to use a hall probe

• Do lots of error propagation

How to make a B fieldRefrigerator magnet?

B fields

Infinite wire

0

70

1 ˆ2

4 10 /

IB

r

T m A

Current loop

Electric dipole

Solenoid

Toroid

0

10

20

30

40

-25 -20 -15 -10 -5 0 5 10 15 20 25

x (cm)

B (

Gau

ss)

a=10 cm

Helmholtz Coils polarized 3He lab at NISThttp:/www.nist.gov

2/32

22

2

2/32

22

20

2 aa xa

a

xa

aNIB

a

a

0

0

0

2

2

0

0045 2

3

xx dx

Bd

dx

dB

a

NIxB

B 25 gauss

SolenoidD0 at Fermilabhttp://www-d0.fnal.gov/solenoid/

Superconducting magnet

B = 2 Tesla (=2x104 gauss)

L=2.7 m, radius = 0.6 mfield uniformity 0.5%

h

l

NIB

NIBh

IsdB

00

00

0

Toroid

G0 magnet at Jefferson Labhttp://www.jlab.org

superconducting toroidB(L) 1.6 T-m

r

NIB

200

Biot-Savart

02

ˆ

4

Idl rdB

r

B field is perpendicular to the current and the vector pointing from the current to the point where the field is being calculated.

rhat is a unit vector pointing from the bit of current dl to the place at which you want to calculate the field.

How can we measure B fields?

Note: only the component of B perpendicular to the current is measured!

Hall EffectWhat is the size of the potential?

Force on the current is zero when the force due to the electric field cancels out the force due to the magnetic field

; w is the width

; t is the thickness

d

d

d

d d

qE qv B

E v B

V wE wv B

I nqv A nqv wt

IBV

nqt

Hall EffectIB

Vnqt

To get something big enough to measure, you want to make I and B as big as possible, t as small as possible.

B=1T, I=10A, t=0.0008 m (thinnest wire that can carry 10A).

For copper, n=5x1028 m-3, q=1.6x10-19 C

628 19

10 11.6 10

5 10 1.6 10 0.0008V V

Pretty small!

Hall Probe Meter

• orientation of probe important

• zeroing of scale important

• calibration of probe important

Experiment

Need to double up on this lab!

Big currents! Use 10 A hole!

Two scales on gauss meter. Upper and lower. When to read each one?

Calibrating the Probe• Calibrate the gauss meter before using it. Some are broken! This is your chance to check!• Set probe to calibrate, read calibration constant on the probe, and adjust screw until you get that reading• Turn to measure and zero using the zero knob

The tips of these probes are very fragile! These probes are very expensive! That’s why you have to double up. Please, please, please, please, please be gentle with them!

hints

• Read your lab manual carefully. Lots of important details in the instructions.• keep your credit cards, etc away from magnet (watches, pdas, memory sticks) • axial probe (round) for loops, transverse (flat with round plastic cover) for toroid•open switch when not actually doing measurement: large currents can overheat the coil.• when push button, allow for settling time• to center, move the probe in and out and look for the position that gives maximum field

hints

•when doing linear fits, taking data so as to give the largest possible range in x values is very helpful!• for toroid, be careful not to go into edges of coil

Systematic Errors

Assume all other errors are small compared to the measurement of the B field.

The manual for the Hall Probe meter gives a 4% systematic error for this device.

In quad

1 + 4 is 5

1 in quad with 4 is 4.1

(prove to yourself that all other measurements have systematic errors that, in quad with 4%, are neglible.)

Single Coil

B vs I at center of coil0

2

N IB

a

Fit B vs I. Verify the intercept is consistent with zero. Extract N from the slope. Take into account the random errors and the syst error on B as you did last week. (random errors go into the fit: systematic error is calculated for the slope, not for the data points, and is added in quad with the error on the slope due to random errors from the fit)

Single Coil

20

2 2 3/ 2

22 2 0

2 ( )

3ln( ) ln( ) ln( )

2 2

N I aB

a x

NIaB a x

2 2ln( ) and v=ln(B)u x a

B vs x at fixed I (read carefully in your lab manual about what happens if you don’t have your center position quite right)

203

ln( )2 2

NIav u

Fit u vs v. Verify slope is consistent with 3/2. Will extract N from intercept. But, need random errors in u,v to do fit.

Single Coil

ln( )

1

Bv

v B

v

B B

B

Single Coil

2 2

2 2

2 2

2 2 2 2 22 2 2

ln( )

2

2

1(4 4 )

( )u x a

u x a

u x

x x au a

a x a

x ax a

More useful math, cont2

2

2 22

ln( )2

2

2, ,

2

b

I aN b

NIab

N eIa

N N N N NN

b I I a a

NI a

Single coil

Add random errors on the intercept from fit in quad with sys error from B field. (translate sys error in B field into sys error on v and propogate that into a sys error on the intercept using the formulas from last week.)

Then, calculate N and the error on N

Two coils

skip

Toroid

Do toroid

errors

See magnet-errors.doc on “lectures” link on web page

Syst error: B field dominates: can ignore rest compared to that (one of the beauties of adding in quad)