Problems And Numericals On Electrostatics Gauss Law And Potential

Capacitors 1. Use the fact that the field due to a plane of charge is E = σ/2ε0 and the Principle of Conservation of Charge to prove the following. (a) For very large thin conducting plate with total charge Q and area A, show that the charge and charge densities on the two sides are equal to ½Q. (b) Consider two very large conducting plates which are initially far apart. One has total charge Q1 and the other has total charge Q2. Show that when the plates are brought closer together so that only a small gap separates them, that the charge on each plate redistributes itself so that the sides facing the gap have equal and opposite charges and charge densities while the outer sides must have equal charges and charge densities. Determine the net electric field to the left, between, and to the right of the plates. 2. Two very large thin square conducting plates are very far apart. The area of the plates is A. A total charge of +Q is added to the first plate; −Q to the second. The two plates are now brought closer together so that only a small gap separates them. (a) Use the results of question 1 to find the charge on each side of each plate. (b) Determine the electric field (i) just to the left of the plates, (ii) between the plates, and (iii) just to the right of the plates. 3. Two very large thin square conducting plates are very far apart. The area of the plates is 5.0 m2. A total charge of 7.0 μC is added to the first plate; −5.0 μC to the second. The two plates are now brought closer together so that only a small gap separates them. (a) Use the results of question 1 to find the charge on each side of each plate. (b) Determine the electric field (i) just to the left of the plates, (ii) between the plates, and (iii) just to the right of the plates. 4. Two capacitors, one with charge QA and the other with charge QB, are connected by a single wire as shown in the diagram below. Prove that the charge on the plates does not change when the switch S is closed. Use the results of questions 1 and 2.

5. Two capacitors are charged separately to the same potential V. They are then connected in parallel with positive plate to positive plate and negative to negative.

Prove (a) that the charge on the each capacitor resides only on the inner faces of the plates and (b) that the charge on the capacitors does not change when the switch S is closed. Use the results of questions 1 and 2.

6. A capacitor consists of two parallel flat plates of area A with equal and opposite charge density σ and separation d. The electric field between the plates is constant at E = σ/ε0 and is directed from the positive plate to the negative. Find an expression for the capacitance of the parallel plate capacitor. 7. A capacitor consists of two concentric spherical shells. The inner shell at radius R has positive charge Q. The outer shell at radius R + d has charge −Q. The distance d is much smaller than R. As a result the electric field is radial and has a magnitude E(r) = Q/4πε0r2. Find an expression for the capacitance. 8. A 4.00 μF capacitor and an 8.00 μF capacitor are separately charged by a 20.0 Volt power supply. The capacitors are then placed in the circuit shown below. (a) What is the charge on each plate when both switches are open? (b) What will be the charge on each plate when both switches are closed?

9. A 4.00 μF capacitor and an 8.00 μF capacitor are separately charged by a 20.0 Volt power supply. The capacitors are then placed in the circuit shown below. What will be the charge on each plate when both switches are closed? (Note that the polarities are switched from the previous question.)

10. In the figure below, C1 = C5 = 3.00 μF and C2 = C3 = C4 = 2.00 μF. What is the equivalent capacitance of the circuit?

11. What is the equivalent capacitance of the circuit shown below?

12. A capacitor consists of two parallel flat plates of area A with equal and opposite charge density σ and separation d. The electric field between the plates is constant at E = σ/ε0 and is directed from the positive plate to the negative. A dielectric of thickness 1/3d and constant κ is inserted next to the positive plate. Find an expression for the capacitance of the parallel plate capacitor. 13. A capacitor consists of two concentric spherical shells. The inner shell at radius R has positive charge Q. The outer shell at radius R + d has charge −Q. The distance d is much smaller than R. As a result the electric field is radial and has a magnitude E(r) = Q/4πε0r2. A spherical shell dielectric of radius ½d is inserted next to the negative shell. Find an expression for the capacitance. 14. Two oppositely charged conducting plates, with equal magnitude of charge per unit area, are separated by a dielectric 3.00 mm thick, with a dielectric constant of 4.50. The resultant electric field in the dielectric is 1.60 × 106 V/m. Compute (a) the charge per unit area on the conducting plates, and (b) the charge per unit area on the surfaces of the dielectric. 15. Two parallel plates have equal and opposite charges. When the space between the plates is evacuated, the electric field is 3.60 × 106 V/m. When the space is filled with a dielectric, the electric field is 1.20 × 105 V/m. (a) What is the charge density on the surface of the dielectric? (b) What is the dielectric constant?

16. A capacitor that has air between its plates is connected across a potential difference of 12 V and stores 48 μC of charge. It is then disconnected from the source while still charged. (a) Find the capacitance of the capacitor. (b) A piece of Teflon (κ = 2.1) is inserted between the plates. Find the voltage and charge on the capacitor. (c) Find its new capacitance. 17. Determine (a) the capacitance and (b) the maximum voltage that can be applied to a Teflon-filled parallel-plate capacitor having a plate area of 1.75 cm2 and dielectric thickness of 0.04 mm. Breakdown occurs when the electric field exceeds 60 × 106 V/m. 18. In question 10, a potential of 600 V is applied across points A and B. What is the charge on each capacitor? What is the energy stored in each capacitor? 19. In question 11, a potential of 24 V is applied across points A and B. What is the charge on each capacitor? What is the energy stored in each capacitor? 20. Three 10-μF capacitors are connected in parallel. A dielectric κ = 2.0 is inserted into one of the capacitors. The capacitors are then connected to a 4.0 V battery. (a) What is the charge on each capacitor and what is the energy stored by each capacitor? (b) The battery is disconnected. The dielectric is then removed from the capacitor. What is the new charge on each capacitor and what now is the energy stored by each capacitor? (c) What happened to the lost energy?

Gauss' Law 1. Using the integral form of Coulomb's Law to find (a) Find the electric field at a point a distance a from one end of a long thin wire of length L and total charge Q. Examine the limit a > L and show that your result is identical to that of a point charge. (b) If the charge distribution was λ(x) = 5Qe-x/5/[1-e-L/5]. Find the electric field. There is no need to evaluate the integral.

2. Using the integral form of Coulomb's Law, find the electric field at a point a distance a from the centre of a long thin wire of length L and total charge Q. The identity ∫du[u2+v2]3/2 = u/{v2[u2+v2]½} + C may be of use. Show that your result reduces to that of a point charge in the limit a >> L. Also show that your answer reduces to E = 2kλ/a in the limit L >> a, the well-known and very useful result for a long thin wire. Note λ = Q/L.

3. Using the integral form of Coulomb's Law, find the electric field at a point midway between two long thin wires of length L and total charge Q and off to one side a distance h. The distance between the wires is a. The identity ∫du[u2+v2]-3/2 = u/{v2[u2+v2]½} + C may be of use. Examine the limit h >> L.

4. A wire has been bent into a semicircle of radius R. It has a linear charge density λ. Determine the electric field at point P, at the centre of the circle. The identity S = Rθ and dS = Rdθ may help.

5. Three long thin wires of length L and charges Q, Q, and -Q are arranged to form an equilateral triangle. Use the result of question # 2 to find the electric field at the centre of the triangle.

6. The "Gaussian Surface" for an infinitely large charged plate is a pillbox of surface area A and length 2a centred about the origin. The plate has a positive surface charge σ. (a) How much charge is contained in the pillbox?

(b) Symmetry demands that the electric field point in the +x direction on the right side of the plate and -x on the other side. Explain why this is so. (c) How much electric field passes through the sides of the pillbox? (d) The pillbox has a thickness a on either side of the plate. How do the magnitudes of the electric field compare at x = -a and x = +a? (e) Use Gauss's Law to determine the electric field at a. (f) Sketch the electric field E(x) as a function of x.

7. A very long thick plate has a uniform positive volume charge density give by ρ(x) = ρ0 for -½d ≤ x ≤ ½d, where ρ0 is a positive constant, d is the thickness of the plate and x = 0 is the centre of the plate. (a) What is the symmetry of the electric field for this object? (b) What is the electric field at x = 0? Explain why. (c) How much charge is contained in the a "gaussian pillbox" of surface area A and thickness 2a where a < ½d. Use Gauss' Law to find E(a) . (d) How much charge is contained in the a "gaussian pillbox" of surface area A and thickness 2a where a > ½d. Use Gauss' Law to find E(a). (e) Sketch the electric field for all a.

8. The "Gaussian Surface" for cylinders is also a cylinder. It has length L and radius a. Consider an infinitely long wire of uniform charge per unit length λ. (a) How much electric field passes through the ends of the "gaussian cylinder".

Explain. (b) How much charge is contains in the "gaussian cylinder"? (c) Use Gauss's Law to determine the electric field at a distance a.

9. A very long thin cylindrical shell of radius R carries a negative surface charge density σ. (a) If the radius of the "gaussian cylinder" is smaller than R, i.e. a < R, how much charge is contained by the gaussian surface? (b) What, therefore, is the electric field everywhere inside the cylindrical shell? (c) How much charge is contained in the "gaussian cylinder" when a > R? (d) Use Gauss's Law to determine the electric field at a. (e) Sketch the electric field as a function of a. 10. A solid sphere of radius R carries a total positive charge Q uniformly distributed thoughout the sphere. The "Gaussian Surface" for a sphere is a sphere of radius a concentric with the sphere. (a) What is the electric field at the centre of the sphere? Explain? (b) If a < R, how much charge is contained inside the "gaussian sphere"? (c) Use Gauss's Law to determine the electric field at the surface of the "gaussian sphere". (d) If a > R, how much charge is contained inside the "gaussian sphere"? (e) Use Gauss's Law to determine the electric field at the surface of the "gaussian sphere". (f) Sketch the electric field. 11. A spherical shell of inner radius R and outer radius 2R, has a uniform charge distribution and total charge Q. (a) Determine the charge inside the "gaussian sphere" for the three regions i. ii. iii.

) 0 < a < R, ) R < a < 2R, ) 2R < a.

(b) Use Gauss's Law to determine the electric field at the surface of the "gaussian surface" when iv. v. vi.

) 0 < a < R, ) R < a < 2R, ) 2R < a.

(c) Sketch the electric field for all a.

12. Use Gauss's Law to determine the electric field as a function of distance a from the centre of a thin spherical shell of radius R and negative surface charge -σ. Sketch the electric field as a function of a. By looking at the electric field, can we distinguish between a sphere with all its charge of the surface and a sphere with the charge smeared throughout the sphere?

Electrostatic Potential 1. (a) E = 5x2i. Calculate ΔV between x = 0 m and x = 4 m. (b) E = (2/y)j. Calculate ΔV between y = 2 m and y = 5 m. 2. Find the electrostatic potential difference between points A and B which are distances rA = 2.0 m and rB = 1.0 m from an infinitely long thin wire with λ = 1.0 μC/m. The result E = λ/2πε0r is useful. If an electron (q = -e = -1.602 × 10-19 C and mass me = 9.11 × 10-31 kg) is released from rest at point A, what is it's speed at point B? 3. Find the electrostatic potential between points A and B which are distances rA = 2.0 m and rB = 1.0 m from an infinitely large thin plate with σ = 1.0 μC/m2. The result E = σ/2ε0 is useful. If an electron (q = -e = -1.602 × 10-19 C and mass me = 9.11 × 10-31 kg) is released from rest at point A, what is it's speed at point B?. 4. In Gauss? Law question #7, we found the electric field due to a uniformly charge long thick plate to be

where d was the thickness of the plate, ρ 0 was the volume charge density, and a is the distance from the centre of the plate. Determine V(x) relative to V(0) where x is the distance from the centre of the plate. Assume V(0) = 0. Plot V as a function of x. 5. In Gauss? Law question #8, we found the electric field due to a long thin wire of uniform charge density λ to be E(a) = (λ/2πε0a)r where a is the radial distance from the wire and r is the unit radial vector. Determine V(r) relative to V(r=1) where r is the distance from the centre of the cylinder. Assume V(1) = 0. Plot V as a function of r. 6. In Gauss? Law question #9, we found the electric field due to a long thin cylindrical shell of radius R and negative surface charge density σ to be

where the field is radial (i.e. in the r direction). Determine V(r) relative to V(r = 0) where r is the distance from the centre of the cylinder. Assume V(0) = 0. Plot V as a function of r. 7. In Gauss? Law question #10, we found the electric field due to a solid sphere of radius R and total charge Q to be radial and have the form

where the field is radial (i.e. in the r direction). Determine V(r) relative to V(∞) where r is the distance from the centre of the sphere. Assume V(r=∞) = 0. Plot V as a function of r. 8. In Gauss? Law question #11, we found the electric field of a thick spherical shell of total charge Q and inner and outer radii of R and 2R to be

where the field is radial (i.e. in the r direction). Determine V(r) relative to V(∞) where r is the distance from the centre of the sphere. Assume V(∞) = 0. Plot V as a function of r. 9. In Gauss? Law question #12, we found the electric field due to a thin spherical shell of radius R and surface charge −σ to be

where the field is radial (i.e. in the r direction). Determine V(r) relative to V(r=∞) where r is the distance from the centre of the sphere. Assume V(r=∞) = 0. Plot V as a function of r. 10. The electric field due to a large plate is E = σ/2ε0 as we have seen. A capacitor consists of two identical plates with equal but opposite charge distributions. (a) Show that the net electric field is zero outside the plates and Enet = σ/ε0 between the plates. (b) Show that the potential difference between the plates is directly proportional to the separation d of the plates. 11. Two point charges of q1 = 3.0 μC and q2 = 4.0 μC are situated at the opposite corners of a rectangle as shown below. The short side has length L = 0.25 m. Find the total potential at points A and B. If a free particle of charge qf = 1.0 μC and mass M = 15 g has speed vA = 2.50 m/s at point A and it follows the indicated path, what will be its speed at point B?

12. Five charges are arranged as shown below. What is the electrostatic potential energy of each configuration? The separation between charges is L.

(a) (b) 13. A rod is bent into a semi-circular arc of radius R. The rod has a uniform linear charge distribution λ. Find the potential at the centre of the arc, point P. The identity S= Rθ and dS = Rdθ may be of use.

14. A total charge Q is distributed uniformly along a straight rod of length L. Find the potential at a point P at a distance h from the midpoint of the rod. (Hint: ∫[x2+k2]½ dx = ln[x+[x2+k2]½] + C). Use the gradient with respect to h to find the electric field at that point.

15. Three thin rods of glass of length L carry charges uniformly distributed along their lengths. The charges on the three rods are +Q, +Q, and -Q, respectively. The rods are arranged along the sides of an equilateral triangle. What is the electrostatic potential at the midpoint of this triangle? (Hint: Use the result of the question 20-7.)

16. The electric potential over a certain region is given by V = 3x2y-4xz-5y2 volts. Determine the components of the electric field and evaluate at the point (+1,0,+2). 17. Over a certain region of space, the electric potential is V = 5x-3x2y+2yz2. Determine the components of the electric field and evaluate electric field at the point (1,0,-2).

Coulomb's Law and Electric Fields 1. Copper has a density of 8.890 × 103 kg/m3, an atomic number of 29, and an atomic mass of 63.546 g. Determine the number of electrons per cubic centimetre in a chunk of copper. Atomic number is the number of protons, and hence the number of electrons, in a neutral atom. Atomic mass is the mass of one mole (6.0221 × 1023) of the particular atoms in question. 2. There are three identical metal spheres, A, B, and C. Sphere A carries a charge of +5q. Sphere B carries a charge of -q. Sphere C carries no charge. Spheres A and B are touched together then separated. Sphere C is then touched to sphere A and separated from it. Lastly, sphere C is touched to sphere B and separated from it. How much charge ends up on sphere C? 3. Where would you put a positive charge of +1 μC in the diagram below so that the net force on it is zero?

4. What is the net force on charge A in each configuration shown below? The distances are r1 = 12.0 cm and r2 = 20.0 cm.

5. A charged ball of mass m = 0.265 kg and unknown q is hanging by a light thread from a ceiling. A fixed charge Q = +5.00 μC on an insulated stand is brought close to the unknown charge. As a result, the unknown charge hangs at an angle θ = 38.0° to the vertical as shown in the diagram below. The distance between the two charges is r = 22.0 cm. (a) What is the sign of the unknown charge? Explain how you know this. (b) What is the magnitude of the unknown charge?

6. An object has a mass of 215 kg and is located at the surface of the earth (mass = 5.98 × 1024 kg, radius = 6.38 × 106 m). Suppose this object and the earth each have identical charge q. Assuming that the earth's charge is located at the centre of the earth, determine q such that the electrostatic force exactly cancels the gravitational force. 7. Three point charges q1 = -1.00 μC, q2 = 2.00 μC, and q3 = 3.00 μC are placed at the corners of an equilateral triangle of side length L = 0.250 m. Find the magnitude and the direction of the electric field at (a) a point midway between charges q1 and q2, and at (b) the centre of the equilateral triangle.

8. Find the net electric field at point A in the diagram below.

9. A small ball of mass m = 0.015 kg is suspended floating in an electric field of magnitude E = 5000 N/C. (a) If the electric field is pointing straight up into the air, what is the charge on the ball? (b) If E points straight down? 10. A ball of mass m = 0.010 kg and charge q is tied by a very light string to the ceiling. The effects of a uniform electric field E = 5000 N/C has caused the charged ball to move to one side so that the string makes an angle of θ = 37° with the vertical. Draw the free body diagram of the floating ball. Determine the magnitude of the charge q. How can you tell the sign (positive or negative) of the charge?

DC Circuits and Kirchhoff's Rules 1. Use the loop method to find current through each resistor in the circuit shown below.

2. Use the loop method to find current through each resistor in the circuit shown below.

3. Use the branch method to find current through each resistor in the circuit shown below. Find the potential difference between points A and B.

4. Use the branch method to find current through each resistor in the circuit shown below. Find the potential difference between points A and B.

5. An electronic flash attachment for a camera produces a flash by using the energy stored in a 750-µF capacitor. Between flashes, the capacitor recharges through a resistor whose resistance is chosen so that the capacitor recharges with a time constant of 3.0 s. Determine the value of the resistance. 6. A charged capacitor is connected across a 9600-Ω resistor and discharges to 1% of its maximum charge in a time of 8.3 s. What is the capacitance of the capacitor? 7. Ideal capacitors have an infinite internal resistance. Real capacitors only have a very large resistance as charges leak from one plate to the other. If a capacitor of 8.0 µF has an internal resistance of 5.0 × 108 Ω, how long does it take for one-half of its original charge to leak away? 8. Three identical capacitors are connected with a resistor in two different ways. When they are connected as in part a of the drawing, the time constant to charge up this circuit is 0.020 s. What is the time constant when they are connected with the same resistor as in part b?

9. In the circuit shown below, ε = 12.0 V, r = 0.500 Ω, R1 = 5.00 Ω, R2= 10.0 Ω, and C = 250 µF. Initially, the switch S is open. (a) At the instant S is closed, determine the current supplied by the battery. (b) After the switch has be closed for a long time, determine the current supplied by the battery. (c) What is the voltage drop and charge across the capacitor at this later time? (d) The switch is now reopened, how long does it take for the capacitor to lose 80% of its charge.

10. A galvanometer has a coil resistance of 250 Ω and requires a current of 1.5 mA for full-scale deflection. This device is used in an ammeter that has a full-scale deflection of 25.0 mA. What is the value of the shunt resistance? 11. The coil resistor in an ammeter has a resistance which is 100 times larger than the shunt resistor. The galvanometer reads 10.0 mA when the ammeter is used to measure the current in a simple circuit. Unfortunately, the resistor in the simple

circuit has a resistance which is only 5.00 times as large as the shunt resistor. What would be current through the resistor if the ammeter was not in place? 12. A galvanometer with a full-scale deflection of 2000 µA has a coil resistance of 100 Ω. If it is to be used as a voltmeter with a full-scale deflection of 1.5 V, what would be the required multiplier resistance?

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