Electric Forces between Charged Plates

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1 CP.1 Goals of this lab Electric Forces between Charged Plates Overview deterine the force between charged parallel plates easure the perittivity of the vacuu (ε 0 ) In this experient you will easure the force between the plates of a parallel plate capacitor and use your easureents to deterine the value of the vacuu pereability ε 0 that enters into Coulob's law. Accordingly, we need to develop a forula for the force between the plates in ters of geoetrical paraeters and the constant ε 0. We conduct the experient in air, which as a perittivity equal to that of a vacuu to within one part in The capacitor consists of two circular plates, each with area A. If a voltage V is applied across the capacitor the plates receive a charge ±Q. The surface charge density on the plates is ±σ where σ = Q A If the plates were infinite in extent each would produce an electric field of agnitude E = σ ε 0 = Q Aε 0, as illustrated in Figure 1. Figure 1: The electric field ade by (left) a single charged plate and (right) two charged plates Since each plate contributes equally, the total electric field between the plates would be E total = Q Aε 0 The potential difference is V = E total d = d Q Aε 0, where d is the plate separation. Solving for Q yields Q = Aε 0 V d (1)

2 CP. The plates are oppositely charged, so the attractive force F att between the two plates is equal to the electric field produced by one of the plates ties the charge on the other: F att = Q Q = ε 0AV () Aε 0 d where Equation (1) has been used to express Q in ters of the potential difference V. Prelab Question 1: The force of attraction, F, between two charged etal plates is proportional to ε 0 AV. Show that F has the units of Newtons (N). d Prelab Question : If you had two charged plates with twice the diaeter of the lab apparatus, with the sae separation distance and sae V, how would the force between the plates change? What would you have to do to the separation distance between the plates to ake the force between the plates the sae as the lab apparatus? Equation () was derived under the assuption that the plates are infinite in extent. For exaple, the expression E = σ ε0 for the electric field is rigorous only for an infinite uniforly charged sheet. A sall correction factor is needed to account for the finite size of the circular plates. Denoting the diaeter of each plate by D, the corrected forula for the attractive force is F att = ε 0AV d ( 1 + d D) (3) The final factor is the correction ter, which is unity if the diaeter of the plates is very large, but has a easurable effect for plates of finite diaeter. The lab apparatus is illustrated in Figure. The object of the experient is to easure the attractive force between two circular plates and infer the value of ε 0. Figure shows the diagra of the forces and the torques involved. The lower plate is fixed, but the see-saw fored by the upper plate and the ass pan attached to it and the counter-balance weight rotates about the pivot point, A. The distances R and R 0 and the ass of the counter-balance have been chosen so that in the absence of a ass on the ass pan and/or an attractive electric force between the plates there is a Figure : Torque Diagra of the parallel plate apparatus

3 CP.3 counter-clockwise torque τ on the syste. Rotational equilibriu is established by adding additional ass to the ass pan and/or applying a voltage between the plates to produce an attractive force between the. Since both the gravitational force F g and the electrical force F att have a lever ar R with respect to the pivot when the balance ar is level, the equation of rotational equilibriu is τ = RF + RF = Rg + RF g att att Using (3) for F att ε AV = + d τ Rg R (4) d D where V is the specific value of the voltage that produces rotational equilibriu in the presence of the ass. Since it is difficult to deterine the precise value of the voltage that will produce equilibriu, we will adopt the opposite tack and find the sallest voltage that will produce a torque in the clockwise direction, thereby causing the plates to coe together. If the voltage is increased in sall increents, this voltage will be a good approxiation to V. Accordingly we rewrite Equation (4) in the for ε A 1 + gd d V D = 0 τ + Rg (5) Thus, a plot of vsv will be a straight line whose slope is equal to the product of easurable and ε 0. The intercept of this line can be analyzed to deterine τ although we will not in fact do this. Figure 3: The parallel plate apparatus

4 CP.4 Caution: Although the current available fro the high voltage supply is too low to cause any peranent daage, the voltage on the capacitor plates is high enough to cause a distinctly unpleasant sensation if you touch the when the voltage is turned on! Note: There is a high voltage probe connected to the power supply which reduces the voltage read by the eter by a factor of Thus, if the eter indicates 0.05 volts, the actual voltage is = 50 Volts. Questions 1. In Part II step 3) why would you not want to plot V vs.? 1 C. Did your value for ε 0 coe within error of the accepted value of ? N 3. What is the effect of an error in the plate separation distance on your experiental results? 4. What is the reason for the correction factor d D in Equation (3)? How big of an effect does it have? Part I Procedure: Calibrating the balance The balance is sensitive to very sall forces, air currents and vibrations. Avoid touching it or the connecting cable while aking your easureents. Avoid sudden oveents, such as flipping of pages, which ay generate air currents. Don't ove the balance! 1. With the power supply off, easure the diaeter D of the plates. This can also be done after your easureents are ade.. The capacitor plates ust be adjusted so that they are concentric and parallel. Adjust the upper plate until it is concentric with the lower (fixed) plate. While doing this ake sure that the cross rod that pivots the plates is not touching the corners of the square holes in the supports. 3. Now look at the horizontal space between the plates and check to see whether the plates are parallel to each other when the plate separation distance is approxiately 1/16 inch. If they are not, ask the instructor to adjust the. Fro here on avoid any change in the alignent of the plates. 4. Place 500 g on the ass pan. The upper plate should not ove. Add 50 g. The upper plate should now ove downward. If it does not, adjust the counterbalance weight and try this again. 5. If the capacitor plates coe too close together when there is a high voltage between the, an arc will occur, as indicated by a sizzling noise. At the sae tie, you can get a variable reading or coplete blank-out of the volteter. Adjust the lower screw (a) (b) Figure 4: Plate Separation Adjustent with the F bracket (a) showing the side view and (b) showing a perspective view.

5 CP.5 in the F bracket to avoid this arcing so that the upper plate cannot coe closer than 1 to the lower plate by setting the screw to leave a sall space between the plates. When the two screws in the F-bracket are properly adjusted the upper plate can only ove downward a sall distance, about 1/3 inch (1 ). The volteter reading should tell you the correct voltage. 6. To set the plate separation d, insert a sall copper plate with a precision ball bearing of approxiately 1/16 inch diaeter fitted in it, between the plates. Position the ball bearing so that it is located about one third of the way in fro the outer edge of the plate (avoid the center). 7. Set the digital eter to easure DC volts. Set the power supply voltage to 50 volts (recall the eter reading will be volts due to the probe). Adjust the thub-screw until the upper plate just barely touches the bearing. This is indicated by the voltage going to zero, which eans there is electrical contact between the capacitor plates and the ball bearing. Turn the thub-screw counter-clockwise a sall aount and then turn it clockwise again, to find the exact point at which contact is ade. 8. Carefully reove the copper plate and bearing. The plate separation has now been set equal to the diaeter of the precision ball bearing. Measure the diaeter of the bearing with a icroeter. Use the tweezers to discharge any residual charge on the capacitor by touching the lower plate and the base at the sae tie. Part II: Acquiring data 1. To a good approxiation the voltage which causes the upper plate to ove downward equals the voltage V in Equation (5). Be sure that the weights are placed at the center of the ass pan. Find the voltages that will ove the upper plate with no ass on it and then with 500 g on it. This will define the range of voltages you will be graphing.. Take data for the interediate asses. For each ass decrease the voltage to zero and then slowly increase it until the upper plate oves abruptly toward the lower plate. (You will hear a soft click when this happens.) The reading on the volteter when this occurs is an approxiate value of V. To refine the easureent, lower the voltage and repeat this procedure four or five ties. Find the best value of V. The uncertainty should not be greater than a few percent. 3. Using the relationship between ass and voltage in Equation (5), ake a graph and deterine the slope to find ε 0 with its uncertainty. JB 7/15/005 electric forces.doc

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