CYK\2009\PH102\Tutorial 7 Physics II 1. A linear inhomogeneous dielectric is sandwiched between the plates of a parallel plate capacitor (separation between the plates a distance y = d) charged to the charge density K The permittivity of the dielectric at y d = 0 1 + K where σ. from one of the plates, is given by is a positive constant. (Neglect edge eects.) (a) Find the expressions for E, D, and (b) Find the bound charge densities σb P. Plot these quantities as a function of and ρb . y. ρb . Plot (c) Find the potential dierence between the plates. Solution (a) First note that the E, D and P are functions σ . Using Gauss law, of y only and are in density on the plates be D(y) = σ E(y) = D(y)/(y) = 0 σ 1+K y d σK yd P (y) = D − 0 E = 1 + K yd (b) The bound surface charge density σb (y = 0) = σb (y = d) = P · (−ŷ)|y=0 = 0 σK P · (ŷ)|y=d = 1+K The bound volume charge density d P dy σK 1 = − d 1+K ρb (y) = − (c) The potential dierence ˆ V = − y 2 d d E dy 0 = − The limiting value, as K → 0, is −σd/0 , σd ln (1 + K) 0 K as expected. Ε0 E P D Ρb d d 1 ŷ direction. Let the charge 2. A current is owing in a thick wire of radius current density at a distance r a. The current is distributed in the wire such that the from the axis is given by r2 1+ 2 a J = J0 ! . Find the total current through the wire. Solution: Take a cross section of the wire that is perpendicular to the axis. Then, net current is ˆ J · dS I = S ˆ a ˆ 2π J0 = 0 = 0 r2 1+ 2 a ! dr(rdφ) 3 πJ0 a2 2 3. Consider a wire, bent in a shape of a parabola, kept in XY plane with focus at origin. The disntace from apex to focus is d. I. The wire carries current Find the magnetic eld at origin. Solution: Equation of the parabola: Let r = rr̂ r (1 + cos θ) = 2d dl = Then, with θ : −π → π be the vector pointing to the parabola. The dierntial tangent vector is given by I dl × (0 − rr̂) = Ir2 dθẑ. B(0) = = = = d r dθ = dθ dr dr̂ +r dθ dθ dθ dr = r̂ + rθ̂ dθ dθ r̂ Then the magnetic eld at origin ˆ µ0 π Ir2 ẑ dθ 4π −π r3 ˆ π µ0 Iẑ 1 dθ 4π −π r ˆ µ0 Iẑ π (1 + cos θ) dθ 8πd −π µ0 I ẑ 4d r Θ 4. [G5.44] Use the Biot-Savart law to nd the eld inside and outside an innitely long solenoid of radius R, with n turns per unit length, carrying a steady current and Eq 5.39. Do z -integration rst.] 2 I. [Write down the surface current density 5. Consider a circular ring, of radius R and carrying current I is placed in the XY plane with its center at origin. Set up the integral to nd the magnetic eld at a point on the X axis, at a distance from the origin. Now, expand the integrand in the powers of the eld. Express in terms of m = I(πR2 ). Solution: 3 R/d d ( R) and nd the rst non-zero term of 0 • r = dx̂ • r = Rŝ = R (cos φx̂ + sin φŷ) 0 1/2 0 Θ • r − r = d2 + R2 − 2Rd cos φ r¢ • dl = Rdφ φ̂ = Rdφ (− sin φx̂ + cos φŷ) r Then the magnetic eld at 0 r • Idl × r − r = IR (R − d cos φ) dφẑ is 0 B(r ) = µ0 4π ˆ 2π IR (R − d cos φ) dφẑ (d2 0 + R2 − 2Rd cos φ)3/2 Now use, " 1 3 |r0 − r| = #3 1 (d2 + R2 − 2Rd cos φ)1/2 " #3 " = ∞ n R 1 X Pn (cos φ) 3 d n=0 d ≈ 1 R 1 + 3 cos φ + O d3 d R d 2 !# Then, ˆ µ0 IRẑ 2π R R B(r ) = − cos φ 1 + 3 cos φ dφ 4πd2 0 d d ˆ 2π R µ0 IRẑ 2 − cos φ + ≈ 1 − 3 cos φ dφ 4πd2 0 d µ0 IπR2 ẑ = − 4πd3 0 This is consistent with the dipole eld in the direction perpendicular to the direction of the dipole. 6. [G5.6] (a) A phonograph record carries a uniform density of static electricity velocity ω, what is the surface current density (b) A uniformly charged solid sphere, of radius spinning at a constant angular velocity point (r, θ, φ) ω R K at a distance and total charge about the within the sphere. Solution: 4 r z σ. If it rotates at angular from the center? Q, is centered about origin and axis. Find the current density J at any 7. [G5.47] Find the magnetic eld at a point z > R rotating sphere, of problem G5.6 5 on the axis of (a) the rotating disk and (b) the 6