PH504 PROBLEM 1 A solid sphere of radius R = 40cm has a total positive charge of 26 C uniformly distributed throughout its volume. Calculate the magnitude of the electric field at (a) 0 cm, (b) 30 cm, and (c) 60 cm from the centre of the sphere. Solution: (a) From Gauss' law we have E(r) = Q/(40r2) for r > R. For a volume charge density , E(r) = r/(30) for r < R Increases linearly (in proportion) to r. At r = 0 we have E(r) = 0. (b) We have (r) = 26C/Vsphere, where Vsphere = 4R3/3 = 4(0.4m)3/3. This yields (r) = 9.7 10-5 C/m3. We therefore have at r = 0.3 m, E(r) = r/(30) = (9.710-50.3/(38.8510-12))(N/C) = 1.1106 N/C. (c) At r = 0.6 m we are outside the sphere and so have E(r) = Q/(40r2) = (91092.610-5/(0.6)2) N/C = 6.5105 N/C. ******************************************************************************************* PROBLEM 2 A sphere of radius R = 40cm has a total positive charge of 26C uniformly distributed over its surface. Calculate the magnitude of the electric field at 1 (a) 0 cm, (b) 30 cm, and (c) 60 cm from the centre of the sphere. Solution: (a) From Gauss' law we have E(r) = Q/(40r2) for r > R, and E(r) = r/(30) for r < R. At r = 0 we have E(r) = 0. (b) We have (r) = 0 inside the sphere, therefore E(r) = 0 inside the sphere. (c) At r = 0.6 m we have E(r) = Q/(40r2) = (91092.610-5/(0.6)2) N/C = 6.5105 N/C. *********************************************** PROBLEM 3 The results for a conducting sphere applies whether it's solid or hollow. Suppose a sphere holds a charge of -3x10-6 C. Let's look at a hollow sphere, and make it more interesting by adding a point charge at the centre with +5 x10-6 C. What does the electric field look like around this charge inside the hollow sphere? How is the negative charge distributed on the hollow sphere? To find the answers, keep these things in mind: The electric field must be zero inside the solid part of the sphere 2 Outside the solid part of the sphere, you can find the net electric field by adding, as vectors, the electric field from the point charge alone and from the sphere alone We know that the electric field from the point charge is given by q / 0r2. Because the charge is positive, the field points away from the charge. If we took the point charge out of the sphere, the field from the negative charge on the sphere would be zero inside the sphere, and given by Q / 0r2 outside the sphere. The net electric field with the point charge and the charged sphere, then, is the sum of the fields from the point charge alone and from the sphere alone (except inside the solid part of the sphere, where the field must be zero). This is shown in the picture: Take k = 1/ (0). How is the charge distributed on the sphere? The electrons must distribute themselves so the field is zero in the solid part. This means there must be -5 microcoulombs of charge on the inner surface, to stop all the field lines from the +5 microcoulomb point charge. There must then be +2 microcoulombs of charge on the outer surface of the sphere, to give a net charge of -5+2 = -3 microcoulombs. ************************************************************ 3 PROBLEM 4 Consider the superposition of the fields from two planar charge distributions which are separated by some distance d. Place a uniformly charged plane with charge density at x = +d/2 and a similar plane with charge density - at x = -d/2. Describe the electric field strength and direction. The field due to the upper plane of charge is E1 = +/20 j, x > d/2, E1 = - /20 j, x < d/2. The field due to the lower plane of charge is E2 = /20 j, x < - d/2, E2 = - /20 j, x > -d/2. The total field in the region x < -d/2 is E = E1+E2 = -/20 j +/20 j = 0. Similarly, the total field in the region x > d/2 is zero. The field is ZERO outside! In the region between -d/2 and +d/2 the total field is is E = E1+E2 = -/20 j -/20 j = -/0 j. The fields add to yield a uniform field between the planes, but they precisely cancel outside the planes to give zero net field outside. Between the planes the field points from the positive towards the negative plane. This is the common configuration of a parallel plate capacitor. ************************************************************************************ PROBLEM 5 A surface charge density (x,y) is given by (x,y) = 3x2+4y2-xy C m-2. Calculate the total charge contained within the area bounded by x=0+a, y=0+a. 4 Answer: 25a4/12 Coulombs This result is obtained by integrating the equation twice - with respect to x and y (the order doesn't matter) and using the appropriate limits. . PROBLEM 6 A line charge has a constant charge density Cm-1. It extends along the y-axis from - to +. Calculate the E-field at a distance a along the positive x-axis. Answer: E=/(20a) radially outwards for positive Method 1: Split the line charge up into a series a small point charges. Their contribution to the E-field along the y-axis cancel leaving only a component along the x-axis. You should end up with an integral which involves three related variables (distance along the line charge, radial distance from segment of line charge to point a and an angle). You need to eliminate two of these by expressing them in terms of the third variable. The final integral is much easier if expressed in terms of the angle variable. Method 2: This question is much easier if solved using Gauss's law. PROBLEM 7 Two charges of –Q are placed at positions –a and +a on the x-axis. Find the potential energy of the system when a third charge +Q is placed at a position x on the x-axis. Find any values of x for which the system is in equilibrium and identify the type of equilibrium. Are there any equilibrium configurations corresponding to positions of the +Q charge not on the x-axis? Potential energy for -a <x < a , x< -a, x > a : There is one equilibrium position x=0 which is an unstable position. There are no equilibrium positions away from the x-axis. 5 This question has a complication which is easy to miss in that you have to be very careful concerning the signs of the three terms which contribute to the P.E. For example in the region -a<x<+a the P.E. is given by the equation below. However if you try and apply this to the region x>a then the third term gives the incorrect sign requiring the factor (a-x) in the denominator to be written (x-a). For points away from the x-axis there is always a net force towards this axis due to the two negative charges attracting the positive one. It is not possible to cancel out this component and hence there are no points of equilibrium away from the x-axis. PROBLEM 8 A parallel plate capacitor has plates of area 5 cm2 and separation 0.1 mm. If the capacitor is connected to a battery of voltage 25V calculate the charge on the plates of the capacitor and the energy stored. The capacitor is now disconnected from the battery and a dielectric of relative permittivity 5 is used to completely fill the region between the plates. Calculate the voltage between the plates of the capacitor and the energy stored. Answers: Q = 1.1 nC, U = 1.4x10-8 J V = 5 V, U = 2.8x10-9 J Here the charge on the capacitor remains constant. PROBLEM 9 For the capacitor of question 8 the dielectric is inserted with the capacitor still connected to the battery. Calculate the charge on the plates of the capacitor and the energy stored. Is any energy lost or gained by the battery in this process? Q=5.5nC, U = 6.9x10-8 J 6 Energy stored has increased so energy is lost by the battery. Here the voltage between the plates of the capacitor remains constant. PROBLEM 10 A parallel plate capacitor has plates of area 15cm2 and separation 2mm. Calculate its capacitance. A dielectric plate of thickness 1mm and r=4 is inserted between the plates. Find the new capacitance. (Hint: the capacitor is equivalent to two capacitors in series. For two capacitors in series the reciprocal of the total capacitance is equal to the sum of the reciprocals of the individual capacitances). Answers: C=6.6x10-12F, C=1.1x10-11F You can either treat this as two capacitors in series or find expressions for the E-field in the two regions of the capacitor and then integrate E between the plates to find the potential difference and hence the capacitance. PROBLEM 11 A polythene sheet of r=2.3 is placed in an electric field of 2x106 V/m. Find E, D and P within the polythene and the surface charge density. E=8.7x105 V/m, D = 1.8x10-5 Cm-2, P = 1x10-5 Cm-2= 7 PROBLEM 12 (a) State how the electric field is related to the electrostatic potential. [2 marks] (b) In some region of space, the electrostatic potential takes the following functional form in Cartesian coordinates x, y, and z: V(x,y,z) = 1/x2 + 1/(xy), where the potential is measured in volts and the distances in meters. Find the electric field, E, at the point x = 2 m, y = 2 m, z = 2m. What is the value of curl E? (a) The electric field is the gradient of -V (b) The gradient can be expressed in vector form e.g. V (V /x,V /y,V /dz) V ( 2 1 1 , ,0) x 3 x 2 y xy2 Therefore E (3 / 8,1 / 8,0) Curl E = 0 PROBLEM 13 (a) The electric displacement D , electric field E and polarisation P are related through the equation D 0 E P where eo is the permittivity of free space. State how D and P are related to E in a medium with relative permittivity er. [4 marks] (b) A slab of concrete with relative permittivity er = 4.5 is inserted transverse to an electric field of strength 4.5 x106 V m-1. Find the values of E and D within the concrete (remember to state the appropriate units). D = r0E 8 and (therefore) P = 0(r-1)E The electric field is reduced by a factor r : E = 4.5x106 /er V m-1 = 1 x106 V/m The displacement is continuous across the boundary: D = 8.85 x 10-12 x 4.5x106 C m-2 = 4 x 10-5 C m-2 PROBLEM 14 By applying Gauss' law to a charged conducting sphere of radius r and charge Q, the magnitude of the electric field outside the sphere is found to be E = + Q /(40r2). (a) Determine the electric potential and, hence, derive a formula for the self-capacitance of the sphere. Show that the self-capacitance of the Earth is approximately 710 nF. [12 marks] (b) Consider a capacitor which consists of two conducting spherical shells of radii a and b positioned concentrically. Derive a formula for the capacitance. [8 marks] (c) Show that if b is only slightly larger than a then the capacitance is approximately C = A/h where A is the surface area of either sphere and h is the distance between them. [6 marks] (d) Now consider the Earth-ionosphere system as representing such a single capacitor. Take the ionosphere to be a positive charged layer of the atmosphere, at a height above the surface of 70 kilometres. The surface of the Earth is the negative plate. The potential difference between Earth and ionosphere is about 350,000 V. Calculate the capacitance of this system, the total charge on the capacitor and the energy stored in the system. [14 marks] (a) Integrating the electric field with respect to radial distance between the sphere and infinity yields V = + Q /(40r) 9 The self capacitance is C = Q/V = 40r. For the Earth,substituting = 8.8542x10-12 and r = 6,400 x 103 m F m-1 yields C = 710 x 10-6 F b) Assuming again a charge Q, the potential difference between the spheres is found by integrating the same electric field between the radii a and b. b V E.dr Which yields a V Q 1 1 4o a b Hence the capacitance is Q/V = ab C 4o b a (c) Taking a=R and b=R+h And approximately equal so that h << R,, ab = R2 + 2hR + h2 and b-a = h and realising that A = 4R2 yields to zeroth order in h/R C = A/h ( d) The capacitance is found on substituting b= 6470 km and a = 6400 km Or using (c) C = 1.1 x 10-10 (6.4 106)2 / 7 x 104 = 0.064 F Charge Q = CV = 0.064 x 3.5 x 105 = 2.2 x 104 C Energy stored is Q2 / 2V = C2V/2 = 1.4 kJ 10 F PROBLEM 15 a) State how the electric charge density is related to the electric field and how the electric field is related to the electric potential. [4 marks] (b) A student measures the form of the electric potential, V, in a region of space in which there is zero charge density. He claims that it is described by the equation V = k x4y3z2 within a Cartesian coordinate system (x, y, z) where k is a constant. Explain why this result is incorrect. [6 marks] (a) div 0E = E = – grad V (b) Hence we have At points where the charge density is zero this reduces to Laplace's equation When the charge density is zero we need to show the given equation for V does not satisfy this differential equation. = 12 k x2y3z2 + 2 k x4y3 + 6 k x4yz2 which is not Differentiating yields zero except at the origin. PROBLEM 16 State how the capacitance of a parallel plate capacitor is related to the plate area and plate separation. Determine the capacitance for plates of area 20 cm2 and separation 8.854 mm. Calculate the electric potential between the plates, the electric field in the region between the plates and the energy stored when the charge of the capacitor is 44.27 nC. The Capacitance is C = Ao/d 11 (could be derived simply through Gauss’ Law if not remembered) Where A is the area and d is the separation. On substitute with Yields C = 20 10-4 8.854 10-12 / 8.854 10-3 = 2 10-12 F V = Q/ C = 44.27 10-9 / 2 10-12 V = 22.13 10 3 V E = V/d = 22.13 10 3/ 8.854 10-3 = 2.5 MV/m Energy stored is = 0.5 2 10-12 (22.13 10 3 ) 2 = 4.4 10 -4 J PROBLEM 17 A thin wire is bent into a circular ring of radius R. It carries a charge per unit length of . Therefore, the total charge on the ring is 2R. Use symmetry arguments to determine the direction of the electric field on the axis of the ring. Derive an expression for the magnitude of the electric field at a point on the axis of the ring at a distance d from the centre of the ring. 12 Due to the symmetry of the problem we can immediately conclude that the net electric field on the axis of the ring is directed along this axis (see Figure). Consider a small segment of the ring with charge dQ. The electric field generate on the axis of the ring by this charge dQ is equal to The component of the field directed along the axis of the ring is equal to The total electric field due to the entire ring is thus equal to The problem specifies the charge density of the ring. Using the specified density we conclude that 13 14