PH206: Electromagnetic Theory

Assignment 1

January 30, 2015

1. A conical surface (an empty ice cream cone) carries a uniform surfacecharge density s. The height of the cone is a as is the radius of thetop. Find the potential difference between point P (the vertex) and Q(thecentre of the top).

Figure 1

2. Assume parallel plate capacitor geometry only.

VTG

VBG

top, dtop

back, dback

blg d

Top Gate

Back Gate

(Conducting Plate)

(Bilayer Graphene)

(Conducting Plate)

Figure 2

(a) Find out the induced charge density charge density in bilayer graphenin terms of VTG, VBG, top, back, dtop, dback only

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(b) What will be electric field at top back and inside the bilayer graphenif both the layers have equal charge density? Express in terms ofVTG, VBG, top, back, dtop, dback

(c) Find out the electric field if the two layers of bilayer does not haveequal charge density but with a screening such that

n = n0e/d

Where is the screening length.

3. Two infinite parallel grounded conducting planes are held at a distance aapart. A point charge q is placed in the region between them, a distancex from one plate. Find the force on q. Check that your answer is correctfor special case a and x = a/2.

4. A point charge q is placed at a distance d from a grounded spherical shellof radius a(as shown in the figure 3). Solve for the potential for r >> ausing multipole expansion(do not use the method of image charge).

Figure 3

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5. Consider a wedge shaped region bounded by grounded conducting surfaceintersecting at the origin with an interior angle a together with a linecharge q per unit length located at the point (r0, ) within the wedge.Find out the potential for r > r0 and r < r0. Use the separation ofvariables technique.

Figure 4

6. What charge distribution produces the potential = q4pi0e(r/a)

r (1 +r2a )

7. (a) For each of the charge distribution drawn below, explicitly evaluatethe first three (monopole, dipole, quadrapole)

Figure 5

(b) Write the exact potential and expand to the same order to check yourresult

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8. A dielectric sphere of dielectric constant 2 is place in a uniform electricfield E in a medium having dielectric constant 2 as shown in figure. Solvethe potential inside and outside the sphere using Laplace equation thendetermine the Polarization P and draw the field lines inside and outsidethe sphere for 1 > 2, 2 > 1 and 1 = 2.

Figure 6

9. Two concentric conducting spheres of inner and outer radii a & b respec-tively carry charges Q. The empty space between the spheres is halffilled by a hemispherical shell of dielectric constant /0 as shown in thefigure

Figure 7

(a) Find the electric field everywhere between the spheres

(b) Calculate the surface charge distribution on the inner sphere

(c) Calculate the polarization charge density induced on the surface ofthe dielectric at r = a

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10. Two long coaxial cylindrical metal tubes (inner radius a outer radius b)stand vertically in a tank of dielectric oil(susceptibility e, mass density). The inner one is maintained at potential V w.r.t. the outer one. Towhat height (h) does the oil rise in the space betwen the tubes.

Figure 8

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