2. Feynman makes a remark that matter is usually neutral. If someone. creates around 1% disturbance of a charge imbalance in a human

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1 Physics 102 Electromagnetism Practice questions an problems Tutorial 1 a 2 1. Consier a vector fiel F = (2xz 3 +6y)î)+()6x 2yz)ĵ +(3x 2 z 2 y 2 )ˆk. Prove this is a conservative fiel. Solution: prove the curl is zero. 2. Feynman makes a remark that matter is usually neutral. If someone creates aroun 1% isturbance of a charge imbalance in a human being the electrostatic forces are big enough to lift planet earth. Make a crue calculation to prove or isprove this statement. You will have to setup the problem qualitatively e.g, assume a person weights 60Kg an is compose of 90% water an estimate the force assuming an equal charge is present in earth. You nee to only worry about the orer of magnitue an not the exact values. (Such problems are calle Fermi problems. Fin out what are Fermi problems ) Solution: Water has 2 hyrogen atoms an one oxygen weighing 18 atomic units. 1 mole of water is 18 gms an has molecules. Assume 1% of water weighing 60Kg is charge an make an estimate. 3. Consier a vector fiel C erive from a curl of another vector B. Fin the ivergence of this vector. 4. Consier a vector A = ˆrr n where ˆr = r/r = îx+ĵy+ˆkz. Fin.A. (i) What is the value of the ivergence for n = 1 in a space of 3 imensions

2 an 2 imensions. (ii) What is the value of the ivergence for n = 2 an n = 3 Solution: Hence ivergence of A is A = ˆrr n = îxr n 1 + ĵyr n 1 n 1 + ˆkzr A x x = rn 1 n 2 r + (n 1)xr x r x = x2 + y 2 + z 2 = 2x/2r = x/r x A x x = rn 1 + (n 1)xr n 2 x 2.A = 3r n 1 + (n 1)r n 3 (x 2 + y 2 + z 2 ) = (2 + n)r n 1. (i) 3, 2, 1 (ii) 0, 1/r 4 Page 2

3 5. Two long thin wires carry a linear has a charge ensity λ 1, λ 2 per unit length. Fin the force per unit length ue to wire 1 on wire 2 if they are separate by a istance b Solution: Draw a cyliner as a Gaussian Surface. The flux on the surface of the cyliner is Hence E2πrL = λ 1L ɛ 0 E = λ 1 2πɛ 0 r The charge per unit length of wire 2 is λ 2 hence (F/L) = λ 2λ 1 2πɛ 0 b 6. i) What is the energy store in a capacitor with capacitance C charge to a value Q? ii)a parallel plate capacitor of area A = l w with separation. Evaluate the capacitance an charge at a fixe voltage when the gap if fille with (i) air an (ii)with a ielectric FR-4 ( the light green plastic in PCBs) of ielectric constant ɛ = 4. Solution: Work one in charging w = vq Page 3

4 but V = Q/C. w = Q 0 Qq = Q 2 /2C Work out the converse when Q is fixe V is varie an prove U = (1/2)CV 2. As an exercise erive the energy store in terms of electric fiel betwen the parallel plates of a capacitor 7. A point q charge is place at a istance before a groune metal plane. (a)use the image metho to write the potential (b)solve for the charge ensity an the force on the charge. (c). Suppose there are 2 groune planes separate by a istance an you can place a charge anywhere in-between. Fin the force on the charge. Solution: Suppose the charge is above istance above an XY plane at Z = 0 then there is an image charge q at. Hence the potential is (a) V (x, y, z) = 1 q [ 4πɛ 0 x2 + y 2 + (z ) 2 q x2 + y 2 + (z + ) 2] Page 4

5 This is obvious if you look at the potential for a ipole which is zero at the mi point. (b) We showe in class for a metal all charge ensity is at the surface with fiels normal to the surface. We know E n = σ ɛ 0 ˆn [ ] E V n = n surf ace σ = ɛ 0 V n In this case the surface is at Z = 0 [ ] V σ = ɛ 0 z Z=0 V z = 1 q(z ) [ 4πɛ 0 x2 + y 2 + (z ) + q(z + ) 2 x2 + y 2 + (z + ) 2] at Z = 0 σ = q 2π(x 2 + y ) 3/2 The force is simply ue to the image charge F = 1 4πɛ 0 q 2 (2) 2 ˆk 8. A parallel plate capacitor of area A = l w with separation is fille Page 5

6 partially with a ielectric slab of ielectric constant e 1 fin out what happens when the conenser plates are set to a constant charge Q. (Hint : use the result for energy store in terms of charge an capacitance in a previous problem) Solution: C 0 = ɛ 0lw When the ielectric is inserte to a length say x C = ɛ 1ɛ 0 xw + ɛ 0(l x)w The potential store ( U = 1/2CV 2 ɛ1 ɛ 0 xw = (1/2) + ɛ ) 0(l x)w V 2 There is a force ue to graient of potential across the capacitor. ( ɛ1 ɛ 0 w F = grau = (1/2) F = (1/2)V 2ɛ 0w(ɛ 1 1) + ɛ ) 0 w V 2 This solution is eliberately given at constant voltage. Try oing at constant charge as pose in the problem. Page 6

7 9. Consier a circular sheet of charge with ensity σ per unit area. If the raius of the isc is bfin the fiel at a istance a from the circle. Evaluate the fiel at point a when the sheet is of infinite imensions. Solution: Area of a circle A = πr 2 implies an area element A = 2πrr Consier a small annular region of a circle of mean raius r. The electric fiel ue to this at an axial point a will be E = 2σπrrcosθ 4πɛ 0 l 2 tanθ = r/a = r = asec 2 θθ Substituting these we have cosθ = a/l = l = asecθ E = σ atanθ asec2 θ θ 2ɛ 0 a 2 sec 2 θ E = σ sinθ θ 2ɛ 0 The total fiel is integrate from 0 b E = b 0 sinθ θ 2ɛ 0 the value of θ in this range is 0 π/2 Page 7

8 l q r r b Blue region inicates charge White annulus is a region Consiere for integration Page 8

9 π/2 E sinθ θ = 0 2ɛ 0 = σ 2ɛ 0 [ cosθ] π/2 0 = σ 2ɛ 0 Fin out the solution for finite sheet. infinite fin the fiel. In the limit it is Some of you i this with an algebraic notation an others solve the potential. Any one who gives alternate solutions in Latex will get it poste on the course web page with their name on it. L A TEX is a comman base eitor erive from TEX. As an example a comman $\vec{\nabla}\times \vec{a} = \vec{b}$ where anything between the two $ signs means it is a mathematical expression an the commans will prouce approriate equations. The above comman when compile will prouce A = B You can ownloa a free book like Not So short introucution to Latex. The software is also free. 10. A large parallel plate capacitor has a separation an is lying in the X-Y plane with the top plate positive an the bottom plate negative. A ball of mass m is charge to a value +Q an release from the top plate with a spee v along the x-axis just below the top plate. Solve for the trajectory of the ball if g is acting ownwar along the z-axis. Page 9

10 Solution: Solve this using kinematic equations like projectile motion except there is a g an electric fiel. 11. Arrive at Poisson an Laplace equations from the ifferential form of Gauss law. 12. When charges accelerate they raiate energy. This means electrons moving aroun the nuclei like a planetary motion will not be stable. If one consiers only a static arrangement of an electron an a proton it will not be stable. The text showe some arguments using Gauss law. Can you assume Laplace s equation hols in the region between the two charges an show a conition for stable equilibrium is not satisfie. 13. A ipole of moment p is subject to a uniform external fiel. Fin the force an torque on the ipole. What is the potential energy of a ipole in external fiel. 14. You are staning on a very long highway an watching cars that approach you. When the cars are too far away the hea lights appear to be a single lamp. When they approach a little closer you are able to see the two healights separately. Can you fin out if you can use this to comment on ipole fiels. Page 10

Exam #2, Electrostatics

Exam #2, Electrostatics Prof. Maurik Holtrop Department of Physics PHYS 408 University of New Hampshire March 27 th, 2003 Name: Stuent # NOTE: There are 5 questions. You have until 9 pm to finish. You