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?

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.

Easier to make a magnetic field using a current

B fields

Infinite wire

0

70

1 ˆ2

4 10 /

IB

r

T m A

Current loop

Electric dipole

“magnetic 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

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?

Changes to Manual• When the lab manual says “reverse the direction of the current and see if the

magnetic field reverses”, don’t do it. It’s not a good thing to do to an analogue meter.

• Don’t do 2 coils part (Part B)

• For toroid, ignore the last sentence of C1. Insteead, measure from the inside the windings, through the windings, to outside of the windings. However, realize that only a small portion of the data (inside the windings, but not near the edge) will be useful for testing the equation in C.2.

Calibrating the Probe• Calibrate the gauss meter before using it. Some are broken! This is your chance to check!• Set meter to calibrate, read calibration constant on the probe, and adjust large metal screw until you get that reading• Turn to measure and zero using the black zero knob• flip back and forth between calibrate and measure a few times to verify your calibration is stable.

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• for transverse probe (used for measuring toroid) need to rotate probe to get max b field reading before starting

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• for single coil, if you have the position of the center wrong, instead of a straight line, you will get one that bends back on itself. If you have this, adjust the center position until it is straight.

Systematic Errors

Assume all other systematic 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 negligible.)

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

Be careful! The error on a is really a systematic error, since it is common for all values of u. Don’t include it in the errors you send when you are fitting u vs v!

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