PHYS 28: Midterm Exam October 28, 200, 8:00-9:20 Last name (print): Initials: No calculator or other aids allowed
PHYS 28: Midterm Exam Instructor: B. R. Sutherland Date: October 28, 200 Time: 8:00-9:20am Student ID Number: Student Name (print): last first middle Student Signature: (/5) 2 (/5) 3 (/5) 4 (/5) TOTAL (/60)
. A charge q is uniformly distributed on the surface of a hemisphere of radius R situated above the x-y plane. That is, the surface is represented by S = { (Rsinθcosφ,Rsinθsinφ,Rcosθ) 0 θ π/2, 0 φ < 2π}. What is the electric field at the origin? (Express your answer in terms of q, ǫ 0 and R.)
2. Within the volume of an infinitely long cylindrical pipe of radius R centred about the x-axis a charge is uniformly distributed with charge density ρ 0. Use Gauss Law to find the electric field within the pipe. (Express your answer in terms of ρ 0, ǫ 0, and the radial distance r from the x-axis.) NOTE: You need only perform this calculation for r R.
3. The electric field within a uniformly charged sphere of radius R centred at the origin is E = q r R 3ˆr where q is the charge and r R is the distance from the origin. Assuming V = 0 at the origin, find the corresponding electric potential, V(r), within the sphere. (Express your answer in terms of q, ǫ 0, R and r.) NOTE: You need only perform this calculation for r R.
4. A spherical drop of mercury of radius a has capacitance C. This merges with an identical drop of mercury. What is the capacitance of the resulting larger spherical drop? (Express your answer in terms of C.) NOTE: Your answer should not depend explicitly upon the radius a of the drop.
If necessary, use this page for additional calculations.
PHYS 28: Electricity and Magnetism Midterm Exam Formula Sheet Parametric Representation of Curves, Surfaces and Volumes and their Differentials. Curves: C = { r(u) u u u 2 }, d r = d r du du (d r = ˆtds, with ˆt the unit tangent along the curve) a) Line on x-axis between points x = a and b: C = {( x,0,0) a x b} d r = (,0,0)d x ds = d x, ˆt = (,0,0) ˆx b) Circle of radius R in x-y plane centred at origin: C = {(Rcosθ,Rsinθ,0) 0 θ < 2π} d r = ( Rsinθ,Rcosθ,0) dθ = ( sinθ,cosθ,0) Rdθ ds = Rdθ, ˆt = ( sinθ,cosθ,0) ds d r = d r du du 2. Surfaces: S = { r(u,v) u u u 2, v v v 2 }, d S = r u r v dudv, ds d S = r u r v dudv, (d S = ˆndS with ˆn the unit normal to the surface. For a closed surface, ensure that ˆn is outward.) a) Rectangle in x-y plane bounded by lines x = a, x = b, y = c and y = d: S = {( x,ỹ,0) a x b, c ỹ d} ds = (,0,0) (0,,0) d xdỹ = (0,0,) d xdỹ = ẑ d xdỹ ds = d xdỹ, ˆn = (,0,0) ẑ b) Circular disk of radius R in x-y plane centred at origin: S = {(rcosθ,rsinθ,0) 0 r R, 0 θ < 2π} ds = (cosθ,sinθ,0) ( rsinθ,rcosθ,0) drdθ = (0,0,r) drdθ = (0,0,) rdθdr ds = rdθdr, ˆn = (0,0,) ẑ c) Sheath of cylinder of radius R and length H centered about z-axis above origin: S = {(Rcosθ,Rsinθ, z) 0 θ < 2π, 0 z H} ds = ( Rsinθ,Rcosθ,0) (0,0,) dθd z = (Rcosθ,Rsinθ,0) dθd z = (cosθ,sinθ,0) Rdθd z ds = R dθd z, ˆn = (cosθ,sinθ,0) ˆr (where ˆr is radial from axis) d) Spherical shell of radius R centered at origin: S = {(Rsinθcosφ,Rsinθsinφ,Rcosθ) 0 θ π, 0 φ < 2π} d S = (Rcosθcosφ,Rcosθsinφ, Rsinθ) ( Rsinθsinφ,Rsinθcosφ,0) dθdφ = (R 2 sin 2 θcosφ,r 2 sin 2 θsinφ,r 2 sinθcosθ) dθdφ = (sinθcosφ,sinθsinφ,cosθ) R 2 sinθ dθdφ ds = R 2 sinθ dθdφ, ˆn = (sinθcosφ, sinθsinφ, cosθ) ˆr (where ˆr is radial from origin) 3. Volumes: V = { r(u,v,w) u u u 2, v v v 2, w w w 2 }, dv = dsdw (This expression for dv works only if w has units of length (eg w r or z) and is in direction of d S) a) Rectangular block bounded by planes x = a, x = b, y = c, y = d, z = e and z = f: V = {( x,ỹ, z) a x b, c ỹ d,e z f}, dv = dsd z = d xdỹd z (where ds is given in 2a) b) Cylinder of radius R and length H centered about z-axis above origin: V = {(rcosθ,rsinθ, z) 0 r R, 0 θ < 2π, 0 z H}, dv = dsd z = r dθdrd z (where ds is given in 2b) (equivalently dv = dsdr = r dθd zdr where ds is given in 2c with R r) c) Sphere of radius R centered at origin: V = {(rsinθcosφ,rsinθsinφ,rcosθ) 0 θ π, 0 φ < 2π, 0 r R}, dv = dsdr = r 2 sinθ dθdφdr (where ds is given in 2d with R r)
Gradient and Divergence in Different Co-ordinates In the formulae below, f = f( r) is a scalar function and F = F( r) is a vector function evaluated at co-ordinate r f( r) F f F ( ) Cartesian f(x,y,z) F = (Fx,F y,f z ) f x, f y, f F x z x + Fy y + Fz ( ) z Cylindrical f(r,θ,z) F = (Fr,F θ,f z ) f r, f r θ, f z r r (rf r)+ ( ) r Spherical f(r,θ,φ) F = (Fr,F θ,f φ ) f r, f r θ, f rsinθ φ r 2 r (r2 F r )+ rsinθ Fundamental Formulae of Electrostatics F θ θ + Fz z θ (sinθ F θ)+ rsinθ F φ φ. Coulomb s law: Relates the electric field at x to charge(s) at x. Below r x x, r r and ˆr = r/r. (a) field due to point charge, q, at origin: E = ˆr r 2 q (b) field along axis of dipole at origin with moment p = pẑ: E = 2πǫ 0 p z 3 ẑ (c) general formula for field due to continuously distributed charge: E = ˆr dq r 2 on a curve... dq = C... λds where λ is (linear) charge density [charge per length] on a surface... dq = S... σds where σ is (surface) charge density [charge per area] in a volume... dq = V... ρdv where ρ is charge density [charge per volume] 2. Relation of electric potential at x to charge(s) at x ( r, dq, etc defined as in Coulomb s Law above): (a) potential due to a point charge, q: V = q r (b) potential due to a dipole with moment, p = pẑ: V = pcosθ r 2 (c) general formula for potential due to continuously distributed charge: V = r dq 3. Gauss Law (a different statement of Coulomb s law): Relates the charge within a volume enclosed by a surface to the electric field normally crossing the surface. ǫ 0 q enc = E ds where q enc = ρdv S V (a) cylindrical geometry: S E ds = 2πrhE(r) (b) spherical geometry: S E ds = 4πr 2 E(r) where r is radial from axis, h is cylinder length where r is radial from origin Note, Gauss Law gives the law of conservation of charge: E = ρ/ǫ 0. 4. Relations between electric potential V and electric field E: V = E d r and E = V For a test charge, q, the force acting on it is F = q E and its potential energy is U = qv. Note, these relations follow from energy conservation: E = 0 5. Capacitance: C = q/ V (a) parallel plate: C = ǫ 0 A/d (area A, separated by d) (b) cylindrical: C = ǫ 0 2πL/ln(b/a) (length L, inner radius a, outer radius b) (c) spherical: C = ǫ 0 4πab/(b a) (inner radius a, outer radius b) (d) C parallel = C +C 2 +... C serial = C + C 2 +... C