The Charge to Mass Ratio (e/m) Ratio of the Electron. NOTE: You will make several sketches of magnetic fields during the lab.

Transcription

1 The Charge to Mass Ratio (e/m) Ratio of the Electron NOTE: You will make several sketches of magnetic fields during the lab. Remember to include these sketches in your lab notebook as they will be part of your participation grade. 1

2 The Charge to Mass Ratio (e/m) Ratio of the Electron This eperiment has two parts: 1. Map the magnetic field due to Bar magnet Solenoid Set of Helmholtz coils 2. Measure the e/m ratio of the electron. First a little about magnetic fields: A magnet always has a north (N) and south (S) pole. No one has ever observed a monopole, either a north or south pole alone. S S N S N N 2

3 A magnet produces a magnetic field which can be represented by magnetic field lines. 1. Magnetic field lines always form loops. 2. They point away from the north pole and towards the south pole. 3. The number of lines is arbitrary but is proportional to the field strength in a given region of space. Region of strong magnetic field. N S Region of weak magnetic field. 3

4 If a magnet is allowed to move freely in the magnetic field of a second magnet, it will align with the magnetic field lines. N S Basically, a compass will tell you which way the magnetic field lines point. S N A compass is a device that shows the direction of a magnetic field. The needle is basically just a bar magnet that is free to rotate. When it is placed in an eternal magnetic field the N pole will point towards the S pole of the other magnet. 4

5 S N N S The Earth has a magnetic field like that of a bar magnet. Given that the north pole of a compass needle points towards the north geographic pole, which of the following is the correct orientation of the Earth s magnetic field? (a) (b) 5

6 An electrical current in a wire produces a magnetic field. The direction of the field is given by the Right-Hand rule for electrical currents. 1) Put the thumb of your right hand in the direction of the current. 2) Curl your fingers around the wire. The magnetic field lines will wrap around the wire in the direction of your fingers. Wire Wire Out of Page Into Page Into Page Out of Page B B B B I What happens if we reverse the direction of the current? I 6

7 What happens if we bend the straight wire into a loop? I The field lines inside the loop are all going the same direction, pointing out of the page in this eample. 7

8 The magnetic field of the loop looks just like the magnetic field of a bar magnet. N I S This is called an electromagnet. What happens if we put many loops together (with the current going the same direction in each loop)? 8

9 If we put many loops together, we form a solenoid. The total magnetic field of the solenoid is simply the sum of the fields from each loop. I N S Into the page Out of the page Once again, the field just looks like a bar magnet. In this view, the solenoid is cut in half so you can see the direction of the current at the top and bottom of each loop. Notice that the field is very uniform inside the solenoid. 9

10 To measure the e/m ratio we will need a uniform magnetic field, but working inside of a solenoid would be a real pain. So we need someway to produce a uniform field in a region where we have easy access. Consider a thin coil of wire loops with radius R: The magnitude of the magnetic field along the ais of the coil is given by: B ( ) N I NI0 R R The number of loops in the coil. The current in the coil. R B ( ) Permeability of free space: Tm A T = Tesla, the unit of magnetic field 10

11 If we place two such coils a distance R apart, then the region along the ais between them will have a uniform magnetic field. R R The magnetic field around the coils will look something like this. 0 B ( ) R 2 R 2 Two thin coils arranged like this are called Helmholtz Coils. B ( ) NI R R R R 2 R R

12 Since the field along the ais of the coils is very uniform, we can just calculate the magnitude at the center ( = 0). NI R R NI 2R B(0) 2 2 Substitute in: K R R 4 R R 4 5R 4 B(0) R NI Tm A Tm A B KN R I Since K, N and R are all constant (for a given set of coils) the magnetic field B inside the Helmholtz coils is proportional to the current I. 12

13 The magnetic field sensor is a Hall effect probe. The Hall Effect Probe measures the transverse separation of positive and negative charges due to an eternal magnetic field. The probe must be oriented properly with the eternal field to get an accurate reading. Turn off the power supply to the coils and press CTRL-0 to zero the probe. Place the probe in the magnetic field and rotate it until the output has the maimum positive or negative value. Warning: The Hall effect probe is not accurate over 55 G! 13

14 Part One of the Lab 1. Use a compass to measure the magnetic field of a bar magnet. 2. Use a compass to measure the magnetic field of a solenoid. 3. Use a compass to measure the magnetic field of a set of Helmholtz coils. 4. Verify that the field inside the Helmholtz coils is uniform. 5. Measure the intensity of the magnetic field B inside the Helmholtz coils as a function of the current I. Plot B versus I to find the slope. 14

15 A magnetic field eerts a force on a moving electric charge. F qvb q electric charge (Coulombs) v velocity vector of moving charge (m s) B magnetic field vector (tesla T) At each point in the magnetic field there is a field vector that points along the field lines from north to south. This type of vector equation is called a crossproduct. It tells us that the result (F) is perpendicular to the two vectors that make up the product (v and B). F B The direction of F is determined using the Right-Hand Rule. v 15

16 Right-Hand Rule for Magnetic Force 1) Put your flat (right) hand in the direction of the velocity v. v 2) Turn your fingers in the direction of the magnetic field B. F B 3) Your thumb will point in the direction of the force F. v Cutnell and Johnson B 16

17 17 Here is an eample with the magnetic field into the screen. v F q Notice that the magnetic force is a centripetal force, always pointing towards the center of the circle. B r

18 18 What happens if the charge is negative? v F q The direction of the force is reversed at each point. B q F v B Another way to think of it: Use your left hand for negative charges. R L L R L R L R

19 Right-Hand Rule for Magnetic Force + Use your right hand for positive (+) charges. Use your left hand for negative (-) charges. Electrons are negative particles! - Ok, now for some equations. In terms of magnitude, we can write the magnetic force equation as: Where is the angle between the magnetic field B and the velocity v. If = 90 then: F qvb We also know that the force is centripetal, so that: F mv r 2 F qvb sin Since we have two epressions for the force, we can set them equal to one another. 2 qvb mv r 19

20 We can change this equation around to get the mass and charge on the same side: q m This equation will allow us to determine the charge to mass ratio (q/m) of a particle provided we know the velocity (v), the strength of the magnetic field (B) and the radius of the circular path (r). The velocity of the particle may be determined using the conservation of energy. v Br K1 U1 K2 U2 Assume the particle starts from rest and is accelerated through a voltage V. K1 0 U qv qv 2 mv 0 q v 2V m K U mv 2

21 Now our equation for the charge to mass ratio becomes: q v 1 q 2V m Br Br m m Br 2 q 2V q 2 m Square both sides: q m 2V Br 2 One final touch, for an electron: q The units work out if: V is in Volts B is in Teslas r is in meters e m e 2V Br 2 We will measure the magnetic field B in units of Gauss (G), but the equation requires Teslas (T), so use the conversion: T 10 G 19 The charge of an electron: The mass of an electron: 31 e m e C m kg C C kg kg 21

22 The e/m Tube A glass tube is filled with Bulb gas. High speed electrons passing through the gas collide with the gas atoms producing visible light. The filament heats up the gas and strips electrons from their atomic orbits. These electrons are then accelerated using an electric potential (voltage). The magnetic field from the Helmholtz coils causes the electron beam to bend into a circle. The diameter of the circle may be measured using the mirrored scale behind the e/m tube. Ammeter A 5 CURRENT ADJ. e/m Apparatus FOCUS 22

23 The e/m Ratio 1. Adjust the voltage and Helmholtz coil current to bend the electron beam into a circle. 2. Measure the diameter of the electron beam using the mirrored scale. 3. Repeat the first two steps for at least 16 data points. 4. Calculate the e/m ratio for each data point. 5. Calculate the average e/m ratio from all of your values. 23

24 NOTE: You will make several sketches of magnetic fields during the lab. Remember to include these sketches in your lab notebook as they will be part of your participation grade. 24