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Name: Section: ______________________________
Section Instructor: ______________________________ Assignment 2: Electric Potential and Capacitance
Concepts: electric potential energy potential due to point charges potential of charged isolated conductor parallel-plate capacitor capacitor with a dielectric electric potential equipotential surfaces capacitance energy stored in a capacitor & energy density
Reading in text: Chapter 24: Sections 1-7, 10-12. Chapter 25: Sections 1, 2, 3abc, 4-7. Assignment: Solutions posted on Wednesday, February 7. Quiz may be given on or after Tuesday, February 13.
2 1. In an electron gun of an oscilloscope or TV set, an electron starts from rest at the surface of electrode 1 and is accelerated towards 1 electrode 2, passing through a small hole there. The electrodes are connected to a voltage source that keeps a constant potential difference V (usually referred to as just V ) between them. (a) Label the terminals of the voltage source + and -. Which electrode is at a higher electric potential? (b) If the distance between the electrodes was increased while keeping V fixed, how would the electric field between the electrodes change? Please explain. voltage source V (c) What effect would the change described in the previous part have on the final speed of an electron passing through the hole in electrode 2? Please explain.
2. A spherical metal shell with inner radius b and outer radius c has a net charge of + Q . (a) Write an expression for the electric field E r as a function of distance r from the shell's center for each of the three regions: r > c , b < r < c , and r < b. Let positve and negative values of E r refer to directions away from and c towards the shell's center, respectively. (b) Draw a graph of the electric field E r as a function of b distance r from the shell's center from r = 0 to r = 3c . (c) Write an expression for the electric potential V as a function of distance r from the shell's center for each of the three regions: r > c , b < r < c , and r < b. Take V = 0 at infinity. (d) Draw a graph of the electric potential V as a function of distance r from the shell's center from r = 0 to r = 3c . Now a separate point charge -2Q is placed at the center of the shell's hollow cavity, electrically insulated from the shell itself. (e) Draw electric field lines everywhere there is an electric field. (f) Repeat parts (a) through (d) for this case.
Physics 208, Spring 2007
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Cornell University
3. The 6 nm thick membrane of a living cell is highly permeable to potassium ions ( K + ) but is impermeable to large negatively charged protein molecules ( A - ) that are inside the cell. A higher concentration of K + inside the cell compared to the medium outside drives diffusion of K + out through the membrane to produce a charge imbalance and an electric potential difference across the membrane. This membrane potential difference impedes further net flow of K + , setting up electrochemical equilibrium, with the concentrations of K + ions inside k B T c out and out, c in and c out , related by the Nernst equation: V N = ln . Here T is the e c in . . Kelvin temperature, k B = 138 10 -23 J / K is the Boltzmann constant, e = 160 10 -19 C is the elementary charge, and V N is the equilibrium membrane potential difference (the Nernst potential). measuring By V N , the ratio of K + concentrations can be found. Using two tiny electrical probes, one inserted into the cell and another on the outside, the cell interior is measured to be at a potential of V N = -90 mV relative to the outside. The electrolyte interior of the cell is an equipotential region, as is the medium outside. There is no significant separated charge within the membrane itself. (a) In a cross-sectional sketch of the cell and/or cell membrane, identify all locations where an electric field exists. What is the direction of this field? Draw electric field lines to illustrate. Please explain your reasoning. (b) What is the electric field magnitude? Note any approximations that you use. How does this value compare with the electric field needed to ionize air ( 3 10 6 N / C)? (c) Where does separated electric charge reside? Label its sign(s) on your sketch. 4 k (d) Using the expression E = , find the surface charge density . Here = (e) What is the direction of the electric force on a K + ion when it is in the membrane? How does this maintain equilibrium? (f) What is the ratio of K + concentrations c in c out inside and outside the cell? (Use T = 310 K .) 4. Capacitance of a parallel-plate capacitor: (a) Use the expression E = 4 k = 0 to find the electric field magnitude E in the empty space between two identical parallel metal plates with charges +Q and -Q . Let A be the area of the side of each plate that faces the other plate and assume that the spacing d between the plates is small compared to their size. Assume that charges + Q and -Q are uniformly distributed over the plate surfaces that face each other. (b) Next, express the electric potential difference V+ - V- of the plates in terms of E and d , and then use this with your result from part (a) to write V+ - V- directly in terms of Q . (c) Finally, use this result to find the capacitance C of these plates defined as C = Q (V+ - V - ) . How does this compare with equation 25-9 in your textbook? 5. Textbook capacitor: Suppose that you made a capacitor using one page from your physics textbook as insulating material to completely fill the space between two flat metal plates. (a) How much capacitance would you have? (See Table 25-1 in your textbook.) (b) What maximum voltage could you apply across the capacitor without sparking from one plate to the other?
Physics 208, Spring 2007
m = 7.2, is the membrane's dielectric constant.
m
m 0
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Cornell University
(c) Suppose that your neighbor made a capacitor of identical size, shape, and plate spacing but with a gap filled mostly with air. Which capacitor, yours or theirs, would have the greater capacitance and by what factor? (d) Suppose that both capacitors were charged to the same battery voltage . How would the electric fields between the plates in the two capacitors compare? Please explain. (e) Which capacitor would have the greater amount of separated charge stored in it (from the battery circuit) and by what factor? Text questions and problems for study and review: Chapter 24: Questions 1, 2, 3, 4, 5, 8, 9, 10. Problems 1, 4, 5, 7, 11, 12, 13, 14, 17, 30, 31, 33, 37, 40, 46, 52, 53, 54, 57, 58. Chapter 25: Questions 1, 2, 3, 4, 5, 7, 8, 10, 11. Problems 1, 4, 8, 9, 10, 18, 24, 26, 31, 34, 38, 42, 43, 82, 83.
Physics 208, Spring 2007
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Cornell University