Download C- PHYS102 - LAB 3 - eOver-M

PHYS-­‐102 LAB-­‐03 Motion of charges in E and B Fields 1. Objective The objectives of this experiment are to demonstrate the deflection of an electron moving in a uniform external magnetic field and to measure the ratio e/m, the electronic charge to mass ratio. 2. Theory 2.1 The Lorentz Force A stationary charge q experiences a force F in an electric field E according to F = qE. A charge q moving with velocity v experiences a force F in a magnetic field B according to: F = qv ! B x
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[1] The force F is called the Lorentz force. The magnetic field B is measured in units of Tesla (T). The direction of F can be determined by first finding the direction of v ! B using the right-­‐hand rule. The direction of F is the same as the direction of v ! B for a positive charge and opposite to that of v ! B for a negative charge. Also notice that the force in eq.[1] is perpendicular to both the instantaneous velocity and the magnetic field. This means that the magnetic field does no work on the charge – the magnetic field changes the velocity of the charge but not its speed. Furthermore, if the velocity of the charge is perpendicular to the magnetic field, the Lorentz force acts radially inward and the charged particle will move in a circular orbit. Applying Newton’s second law: |qv x B| = qvB = mv2/r where v2/r is the centripetal acceleration. This gives: r = mv/qB [2] Equation (2) can be interpreted as follows: [a] For a given magnetic field B, increasing the velocity of the charged particle increases the radius of curvature r of its orbit. Thus, a fast-­‐moving particle is harder to deflect by the same magnetic field. [b] For a given velocity, as the magnetic field is increased, r, the radius of the circular orbit gets smaller. The charge of an electron is conventionally denoted bye.. Thus for an electron, eq. [2] can be re-­‐expressed as: e/m = v/(rB) [3] This is a ratio of two fundamental constants: the electron’s charge and its mass. In the present experiment you will determine this very important ratio e/m by directly observing the orbital radius r of a beam of electrons of speed v moving in a magnetic field of magnitude B. To determine the electron speed v, we note that the electrons in this experiment are generated via thermal emission from a “cathode” (a wire filament) heated to high temperatures. By heating the filament you are providing enough thermal energy to the electrons for them to “evaporate” from the heated filament. Such electrons are then accelerated from nearly zero speed, through an electric potential difference ΔV before entering the region of uniform magnetic field. As an electron moves through the potential difference ΔV, it acquires a kinetic energy equal to: ½mv2 = e ΔV [4] v = [2eΔV /m]1/2 [5] [6] or From eqs.[3] and [5] one can write: e/m = 2ΔV /(rB)2. 2.2 The Helmholtz Coils To create a region of uniform magnetic field B, we will use an arrangement known as Helmholtz coils (Figure 2). It consists of two circular coaxial coils with N turns each separated by a distance equal to the radius of the coils. The two coils have the same number of turns, carry the same current and in the same direction. This arrangement produces an axial Y
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field that is highly uniform in the region between the coils. The B-­‐field can be expressed as: Bx (x) =
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2 & !(x + R / 2)2 + R 2 # 3/2 !(x % R / 2)2 + R 2 # 3/2 '
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[7] where µo = 4π x 10-­‐7 Tm/A is the permeability of free space, N is the number of turns in each of the pair of coils (N = 130 in this apparatus), I is the coil current and R is the radius of the coil. At x = 0, eq.[7] can be reduced to: Bx (x) =
µo NI ! 4 $
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= 1.168 ' 10 (4 I / R [8] 3. Experimental 3.1 The e/m Apparatus Figure 3.On the left, the e/m apparatus consisting of a pair of Helmholtz coils, vacuum tube and control box containing power supplies for the tube heater, coils and accelerating voltage. On the right, shown schematically, the electron gun consisting of a heated cathode, protective metal grid, anode and power source to the accelerating voltage. The e/m apparatus, depicted in Figure 3, consists of a partially evacuated glass bulb placed between the Helmholtz coils. Inside the glass bulb, electrons are emitted from a heated cathode connected to the negative terminal of a high-­‐voltage power supply. The cathode is partially shielded by a surrounding grid that allows electrons to pass through. The electrons are then attracted to an anode mounted below the grid and connected to the positive terminal of the high-­‐voltage power supply. The electrons escaping through the grid are accelerated downwards towards the anode by an adjustable potential, V. The accelerated electrons emerge from the electron gun with a velocity v and enter a region of uniform magnetic field B. Because the electrons’ velocity is transverse to the B-­‐field direction, the beam is deflected by the Lorentz force F = ev x B resulting in a near vertical circular orbit. 3.2 Experimental Details 3.2.1 Procedure for Collecting Data 1. Set the apparatus on a level table away from sources of magnetic fields – current-­‐carrying coils or any ferromagnetic materials. 2. Optional: To reduce the effect of the earth’s local magnetic field: if you have a compass available use it to determine the direction of the local magnetic North and align the axis of the Helmholtz coils along the direction of the compass needle. If you do not have access to a compass go to the next step. 3. Press the power switch to the ON position. The instrument will begin to go through a self-­‐test taking about 30-­‐sec. During this period all controls are locked out as the system begins to heat the cathode emitter to the proper operating temperature. After the self-­‐test is complete indicated by a “000” on the digital display, wait an additional 5-­‐10 minute warm-­‐up time before starting to collect data. 4. Turn the Voltage Adjust control up to 200eV. While observing the bottom of the electron gun, notice the bluish beam traveling straight down to the envelope of the tube (Note: You will be able to see the electron beam better if you dim the room lights). Actually the electrons themselves are invisible. The orbit of the electrons is made visible by collisions between electrons and a small amount of Helium gas inside the bulb. After colliding with electrons, the orbital electrons in the helium gas atoms are excited and, upon returning to ground state, emit photons (light). This process lights up the path taken by the deflected electrons. 5. Increase the beam current by turning the Current Adjust control up. You should be able to observe the circular deflection of the beam. As the current is further increased, the beam will form a complete circle within the envelope. The diameter of the beam can be measured using the internal centimeter scale inside the tube. The scale numbers fluoresce when struck by the electron beam. 6. For an accelerating voltage of 200V, measure the beam diameter for 4 coil current settings. Record these data in your Data Sheet. 7. Repeat Step 6 for two additional accelerating voltages. You can increase the voltage in two steps of 100-­‐125V each. The accelerating voltage must stay below 500 V. 8. Once you finish collecting the data, turn down the current and voltage controls and switch off the power to the apparatus. 9. Measure the inner and outer diameters of the Helmholtz coils at several places. Determine the average radius of the coils, since the coils may not be uniformly round and may also differ slightly in their other physical dimensions such as the coil thickness. These variations in the physical dimensions may be used later to determine whether it has a significant effect on the accuracy of your results. LAB-­‐03 Motion of Charges in E and B Fields Name:_______________________ Sec./Group__________ Date:_____________ 4. Pre-­‐Lab Consider an electron being ejected from the surface of a heated filament at nearly zero speed. [a] Through what potential difference must it be accelerated to achieve a speed of 8.4x106m/s? Any relativistic effects can be neglected. [b] What is the minimum B-­‐field required to cause the electron in part [a] above to move in a circular orbit of radius r = 5.0 cm? LAB-­‐03 Motion of Charges in E and B Fields Name:_______________________ Sec./Group__________ Date:_____________ 5. Data Average Coil Radius R = ________m B = 1.168x10-­‐4 I/R e/m = 2ΔV/(r2B2) Table 1. Accelerating Voltage = 200 V Coil
Current (A)
B-Field (10-3T)
Orbit Diameter, d (m)
Product rB =dB/2
(Tm x 10-3)
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Mean value of rB = ___________ Table 2. Accelerating Voltage = ___________ V Coil
Current (A)
B-Field (10-3T)
Orbit Diameter, d (m)
Product rB =dB/2
(Tm x 10-3)
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Mean value of rB = ___________ Table 3. Accelerating Voltage = ___________ V Coil
Current (A)
B-Field (10-3T)
Orbit Diameter, d (m)
Product rB =dB/2
(Tm x 10-3)
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Mean value of rB = ___________ LAB-­‐03 Motion of Charges in E and B Fields Name:_______________________ Sec./Group__________ Date:_____________ 6.1 Analysis 1. According to eq.(6), rB, the product of the electron path radius times the value of the magnetic field is a constant for a particular setting of the acceleration voltage ΔV. From the average measured value of rB, calculate the value of e/m for each of the three acceleration voltages. For [2] and [3] (in Table 4 below), use other voltages you used in the experiment. Table 4 Accelerating Voltage, ΔV
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Use the values of e/m from the three cases above and determine an average value. 2. Calculate the theoretical value of e/m using: e = 1.602 x 10-­‐19C and m = 9.109 x 10-­‐31 kg. Calculate the percent difference between the experimental and the theoretical values of e/m. Discuss the possible sources of error to explain this difference. 7. JUST FOR THE FUN OF IT What happens when B is not ┴ v? Permanent magnets will be made available to each group. Do not bring these magnets near your credit cards, bus passes, electronics and other items that can be damaged by strong magnetic fields. Do not bring the magnet close to the electronics equipment in this lab. Slowly move the permanent magnet close to the glass bulb while constantly monitoring any changes in the circular orbit of the electron beam. Do not get any closer than about 4 – 6 to the glass bulb. Changing the orientation of the magnet will allow you to create additional (non-­‐axial) components of magnetic field (along the y-­‐ and z-­‐axes). 1. What will be required to produce a spiraling electron beam?