Analyze the Measurements

Here is the lab video link where the data was collected (https://youtu.be/ydPXXJt-Lt0)

The Physics behind the Measurement Analyze the Measurements

The Physics behind the Measurement

We begin by gathering up the basic data, then calibrate the current balance with gravity, and lastly calculate the force on the wires with different currents. Once we have that we explore for the relationship between the force from the currents that is Ampere's law.

Let the scale on which the laser spot falls be y

measured from the equilibrium position of the spot on the scale with no calibration mass and no current.

Let the distance from the scale to the mirror be ds

and the distance from the parallel wires to the mirror be dw

.

Let the separation of the currents be r

and let r0

be the separation when there is no current and no calibration mass to change it.

Let

be the length of the wire segments that are close and parallel.

Let I

be the current which is measured for each y

of laser spot deflection.

Let fB

be the force from the magnetic interaction of the two currents.

Let fG

be the force of gravity from a calibration mass applied to the upper wire.

Let

be the angle of rotation of the frame holding the upper wire and the mirror from its equilibrium position with no current and no mass. The angle of rotation of the laser beam is 2

because of the optics of the law of reflection. (Imagine holding a small mirror and looking at yourself. Rotate the mirror 45 degrees and you will see to your right or your left, 90 degrees from the line to the mirror.)

To find how far the spot moves for a change in direction of the laser beam, imagine a circle of radius \(R)\. Along the circumference of the circle, if \(R)\ is swept all the way around \(2\pi) radians, the end of the line covers 2R

distance in going around. How much would it cover for a small fraction of that full rotation, say the angle A

? It is

A/2=S/2R

S=RA

The distance along the arc is the radius times the angle subtended in radians. When the angle is small that arc on the circle is nearly a straight line tangent to the circle at that point, what we see as the vertical position of the spot when the laser beam rotates through A

. Since A=2

because of how light reflects off a mirror, the vertical position of the spot is

y=2ds

For the same rotation of the frame, the separation of the currents will be how much the upper wire moves up as the frame rotates

r=dw+r0

where r0

is the equilibrium separation (we will return to discuss this point later).

This device is simply a balance with fixed weights and a wire finding an equilibrium It is exactly the same as the pan balance, and the same physics governs its equilibrium. The rotation from equilibrium is proportional to the total force that's applied on one side. A small rotation of the frame is caused by a force such that

f=

Here

(gamma) is a "balance constant" we will find be measuring the angle of rotation for a known force of gravity on the calibration mass m

. Imagine that the frame is rotated to raise the wire. The balance itself acts to pull the wire down with a force of magnitude

fG=

when the angle is up.

We find the balance constant

by observing the deflection for a known force on a test mass

mg=g

The calibration rotation is found from the deflection of the beam from equilibrium when the mass is added to the upper wire with no current on the wire

g=yg/2ds

where yg

is the magnitude of the downward movement of the laser spot when the calibration mass is added.

The choices of sign here are assuming that the scale increases upward and the calibration mass moves the spot downward. The factor of 2 is because this is the movement of the laser beam which is twice the movement of the mirror and frame. Now we know the balance constant from this calibration measurement is

mg=yg/2ds

=2mgds/yg

When there is a magnetic force to move the wire up, the balance seeks an equilibrium where the frame is at a new angle with no net force

0=fBfG

The forces here are magnitudes and the sign accounts for their direction assuming the magnetic force on the wire is up and the restoring gravitational force is down. We have

fB=fG

fB=

We see the light beam is then displaced on the scale

y=2ds

Since =y/(2ds)

it follows that

fB=y/(2ds)

Now let's substitute for the balance constant

we found and discover that several factors cancel out

fB=(2mgds/yg)y/(2ds)

fB=mg(y/yg)

This is an amazingly simple and intuitive result, so you can start here now that you see the physics behind it.

The force on the upper wire is the gravitational calibration force times the ratio of the displacement with the current to the displacement with the gravitational calibration mass.

From Measurement to Understanding

1. What is the distance from the mirror to the vertical meter scale on which the laser spot falls? Label this distance as ds

.What is the distance from the mirror to the pair of parallel wires? Label this distance as dw

.

2. Use these values to find a scale factor from the laser beam motion to the wire separation change. This deflection constant is

a=(dw/2ds)

such that the change in separation of the wires from the equilibrium position is

R=aY

where Y

is the movement of the laser spot. Find a such that R is in cm when Y

on the scale is in cm.

3. Starting at 12:24, review the video and find the deflection of the laser spot in cm from its rest position as far down as it will go, approximately where the two wires are in contact. Use the values for the distances from the pivot point to the scale, and to the wires to find the actual movement of the upper wire down to the point of contact. It's quite small, so don't be surprised to find a value less than 1 mm. Label that as Rc

.

4. Following that we added a 30 mg calibration mass to produce a downward deflection from the gravitational force on the mass.Label this deflection as yg

and use a positive number because the sign has been accounted for with the math above. The force that produced this deflection is fG=mg. The mass of 30 milligrams is m=30106kg

This deflection allows you to calibrate the movement of the scale in force units. For the following you will find it helpful to measure distances on the scale in centimeters. The smaller divisions on the scale are millimeters. Find the conversion factor from distance on the scale relative to the equilibrium position to force in "micro" newtons, 106

N. Using this for the force unit will make bookkeeping your measurements much easier than if you measured in meters and newtons. Find the constant such that

FB=by

where the force FB

of the magnetic interaction on the upper wire is calibrated from the 30 mg mass's weight and given in micro-newtons, and the movement on the scale from the equilibrium position is in centimeters.

5. Review the video starting at 15:35 where the vertical deflections were measured for various currents. Record the position of the laser spot in centimeters when there was no current, and then for all the currents that were tested record the current along with spot position for each one. You will have two lists, one of the currents, and the other of the position of the spot. Make a third list of the change in the spot position by subtracting the position at equilibrium from your measurements. These changes will be the Y

values from which you will find the force for each of the currents I

.

6. Calibrate the data from part 5 by using the deflection constant b

with the known mass you found in step 4. For each one that you measured find the force F

in micron-newtons on the upper wire from

F=bY

7. Each deflection of the laser spot also represents a different separation of the two wires. We cannot measure the absolute separation, but we can estimate it reasonably well from your measurements and a little extra physics. First, the change in the separation for each current is

RI=aY

with the calibration you found above in step 2. This is how much the separation increases from the equilibrium position. We also know from your measurement how far the wires are from being in contact which is the RC

you found in step 3. Clearly we should add RC to RI

, but that will give the separation of the wires from being in contact. It is not the separation of the currents which are on those wires.

These currents initially will be on the surfaces of the wires and within the wire the currents are in the same direction so they "pinch" into fine threads of current by the self-force of the field they generate. The internal currents in the two wires are oppositely directed and repel, so they push out to the far edges of the wire. Consequently, the separation of the currents when the wires are in contact is twice the diameter of the wires. If the wire diameter is D

, we have

R=RI+RC+2D

for the separation of the currents at each current measured. The wire diameter is approximately 2 mm. Find the R

in cm for each current that was run in the experiment. You now have 3 lists: I, F, and R for the current in A, the force in N

, and the wire separation in cm.

8. You'll notice that the separations are nearly constant at first. The small changes in separation are much less than the separation itself, and the increase in the force with increasing current is largely from the dependence of Ampere's law on I2

. Use Desmos to plot a graph of your measurements of the force versus the current. Focus your attention on the lower currents where R

is nearly constant. What does this graph tell you about the dependence of the force on the current?

9. Calculate a force for each current using the R

from this measurement and Ampere's law. Plot those theoretical forces along with your measurements. How do they differ? If those differences are systematic, can you suggest what may cause them?

10.Use one of your measurements from the middle of the range of currents and from it find an experimental value of the permeability of free space 0

. Estimate an error in this measurement by looking what other currents would have given, and the effect of the low precision of measurement of the spot deflection.