magnetic fields and electric current

8/6/2019 Magnetic Fields and Electric Current

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Magnetic Fields and Electric Current

Goals:

To examine the effect that magnetic fields have on moving charges.

To examine the magnetic field produced by a long, straight current-carrying wire.

Equipment:

Phys212 LabKit Module

Magnetic compass

(2) Stackable banana-plug connecting wires

2-m length of insulated wire

Vertical stand, clamp, and horizontal rod

e/m equipment (Helmholtz coil, discharge tube, power supply)

Software: Microsoft Excel

Introduction:

Magnetic fields share many similarities to the electric fields which we have studied in detail, but

there are some critical differences which will explore in this laboratory activity. First of all,

magnetic fields will be measured in units of tesla (T), where 1 tesla = 1 newton per amp-meter(1 T = 1 N / A·m). In the first portion of today’s lab, we will explore the ways in which magneticfields can exert a force on moving charges.

For a charge q moving with velocity v

in the presence of a magnetic field B

, the force exerted

on the charge by the field is given by

BvqF

. (Eq. 1)

Note the cross product in the above calculation: this implies that the direction of the force on the

charged particle will always be perpendicular to both the velocity and the field. We will makefrequent use of the right-hand rule to help us determine the direction of this force.

A magnetic field will also exert a force on a current-carrying wire. This should make sense, as an

electric current is simply the flow of charge. So, very similarly to Eq. 1, we can write the force

exerted by a magnetic field B

on a straight wire of length L carrying a current i as

B LiF

, (Eq. 2)

where the direction of L

is defined to be the direction of the (conventional) current.

In the second portion of today’s lab activity, we will explore how magnetic fields are produced.

We will come to find that magnetic fields are produced by moving charges (and hence, also byelectric currents.) The magnetic field produced by a current or moving charge can be determined

by using two fundamental laws: the Biot-Savart law (which is the magnetic analog of Coulomb’s



law for electricity) and Ampère’s law (which is the magnetic analog of Gauss’ law fo r

electricity).

Much as we can break down a complicated charge distribution into infinitesimal pieces of

charge, and compute their infinitesimal contribution to the electric field some distance away, we

can also break down a complicated current distribution into infinitesimal current elements andcompute their infinitesimal contribution to the magnetic field some distance away. The Biot-

Savart law does exactly that. It states that the magnetic field contribution Bd

due to an

infinitesimal length element sd

carrying an electric current i is given by

3

0

4 R

Rsd i Bd

, (Eq. 3)

where μ0 is a constant of nature known as the permeability of free space (defined to be equal to

μ0 = 4 π × 10−7

T·m / A), and R

is the displacement vector pointing from the current element tothe point at which the field contribution is to be found. The total magnetic field at a point can be

found by integrating Eq. 3 over the current distribution.

Much as Gauss’ law afforded us with a simpler way of calculating the electric field when we

could employ symmetry considerations, Ampère’s law provides us with a simpler method of

finding the magnetic field around a current distribution when that distribution possesses some

symmetries. It states that the integral of the magnetic field component along a closed path,integrated over that path, is proportional to the amount of current enclosed by the loop. In

equation form, it reads

enc0isd B

. (Eq. 4)

To summarize, equations 1 and 2 allow us to determine the force on moving charges (or

currents) due to any magnetic fields present, and equations 3 and 4 allow us to determine thenature of the magnetic fields created by moving charges (or currents).

We will employ all of these ideas today to help us gain a better understanding of magnetism,especially the motion of charged bodies moving in a uniform magnetic field. To begin, we’llrepeat a classic experiment in which we attempt to determine the ratio of the charge on an

electron to its mass, i.e., the ratio e / m for an electron. We will use a beam of electrons whoseenergy (and hence speed) can be controlled by varying a known accelerating voltage. The

electron beam is formed inside a glass sphere containing nitrogen at a residual pressure of

approximately 10−2

torr. When the electrons collide with the nitrogen molecules, they cause the

latter to emit a faint bluish radiation. This glow will emanate from wherever the electrons collidewith the gas molecules, and as such, will give us a visual cue (visible in a darkened room) as to

what path the electrons are following.

The electrons will be emitted from an indirectly heated cathode, and will be accelerated through

a known electrostatic potential difference V after they are emitted. By conservation of energy, the

potential energy lost by the electrons (recall: ΔU = qΔV ) as they move through the accelerating



Name: ____________________________ Date: ______________

Name: ____________________________ Lab Sect.: __________

Name: ____________________________ Lab Instructor: ______________________

Directions:

Lab Activity 1: Measuring the charge-to-mass ratio for electrons

In this first activity, we will use the fact that magnetic fields exert a force on moving charges todetermine the charge-to-mass (e / m) ratio for electrons. If you have not already done so, carefully

read the introduction of this activity to decide what quantities you will need to measure and how

you will need to combine them in order to determine e /m. The apparatus to be used for this

activity is shown below.

This equipment should have already been properly set up for you. If it does not appear to be

connected, please make sure you consult your instructor. Then follow the directions provided in

precise order.

The equipment consists of two separate items as shown in the figure above:

o The e / m apparatus (Helmholtz coil and discharge tube). Note that the equipment

as set up in the lab will have a wooden cover over this to make it easier to see the

glowing electron path.

o The discharge tube power supply.

Helmholtz Coils

Discharge Tube

Helmholtz Coil

Current Control

DC Power Supply

Voltage Current



Make sure that all the knobs on the power supply are turned all the way down(counterclockwise) and the Helmholtz coil current control indicated above is turned all

the way up (clockwise).

Switch the power supply on and wait for a couple of minutes for the cathode to heat up.

Start turning up the left voltage knob on the power supply slowly until you read a voltage

of around 200 V on the digital display. Record the exact value of this voltage.

Accelerating voltage V : ______________________

Look through the window in the wooden cover at the right side of the glass sphere: youshould be able to see a faintly glowing electron beam going vertically downwards.

Turn the current knob on the power supply till the current display reads about 2 A. Themaximum current through the Helmholtz coils is 2 A, so please do not exceed this value.

The electron beam should now be visible as a circle.

You should not need to change any settings on the power supply for the rest of thisactivity. Only adjust the Helmholtz coil’s settings.

Q1. Why do the electrons follow a circular path when current flows through the Helmholtz coil?

Now, start varying the current using the Helmholtz coil current control on the e/m

apparatus. Note that as you decrease the current, the diameter of the circle changes.

You can see that the left side of the circle passes over numbers on a linear scale that glowas the beam hits them. The number that you read is the diameter of the circle in

centimeters.

Q2. Record (in the table provided) the current I through the Helmholtz coils and the diameter d

of the circular electron path. Make sure that you convert the diameter into meters. Repeat this

procedure for 10 different pairs of the current I and the diameter d .

After you’ve taken all your data, switch off everything: first turn the Helmholtz currentcontrol knob on the e / m apparatus all the way up and then turn down the current to zero

using the knob on the power supply. After this, turn down the voltage to zero. Finally,

you can switch off the power supply.Enter the data from the table above into an Excel spreadsheet that:

o Calculates the magnetic field B from your measured current I (refer to Eq. 7 in the

introduction);

o Calculates (1/ r ) from d , where r is the radius of the circle (in meters).

Create a plot of (1/ r ) versus B.



Try to arrange to have the thin, insulated wire run from the switch on the LabKit board vertically

upward to the horizontal pole, where it can be taped or wrapped to keep the wire relativelystraight. From there, the wire should run across the pole and back down to the battery module.

We want to use the “momentary” position of the switch (position A) to let us connect the circuit

for brief moments at a time, while we measure the magnetic field around the wire with amagnetic compass. The conventional current should travel from the positive terminal of thebattery to the central terminal of the switch, from terminal A of the switch it should travel up the

vertical wire segment, and then back to the negative terminal of the battery along the remainder

of the wire. To connect the thin insulated wire to the LabKit board and batteries, you may wantto use the holes on the sides of the posts near the switch and batteries (thread the post up and

down to secure the thin wire). ( Note: to achieve the greatest possible magnetic field, we will use

both batteries connected together in series to generate our electric current.)

Q4. For the wire segment shown below, use the Biot-Savart Law and/or a "right-hand rule" to

qualitatively sketch the magnetic field lines produced by the current traveling up ward through

the vertical segment of the insulated wire.

i

I

side view birds-eye view



Next, use your magnetic compass to confirm your expectations. Remember that thecompass is always subject to the Earth's magnetic field, and is also affected by otherexternal factors such as the proximity of other magnetic objects or electrical currents.

Typical values for the strength of the Earth’s magnetic field are around 5 × 10−5

T. Keep

in mind that magnetic fields obey superposition, so that you can always subtract theeffect of any "background" magnetic fields by carrying out measurements with and

without a current in the wire. Finally, a cautionary note: the painted end of your compass

needle is supposed to be the "North" pole of the needle; however, your compass needlemay be mislabeled! Check to make sure which end of the needle is "North." Now that

you have located "North", slowly move your compass around the wire while the circuit is

not connected. Does the compass point consistently to the "North"? If not, you will need

to move your test set up to a location where stray magnetic fields are less noticeable.



Q5. While viewing your experimental set up from above, complete the following diagram to

show the compass needle direction without and with current in the wire, when the compass isheld midway up the wire at the four positions depicted.

Q6. By your observations in Q5, and predictions from Q4, how do the magnitudes of the field

produced by the current in the wire and the Earth’s magnetic field compare; is one much larger

than the other or are they comparable; which is larger? (The Earth’s magnetic field is

approximately 5 × 10−5

T). Do your results of Q5 confirm your expectations for the magneticfield produced by the current in the wire? Explain clearly why or why not.

Q7. What do you expect will happen to the magnetic field lines if the current direction is

reversed? Justify this using:

(a) the right-hand rule (thumb in the direction of current version)

Without current in wire With current flowing up wire

I

magnetic fields and electric current

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