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Electrostatics

1. A conducting rod of radius R1 = 1.57mm and length l=13.9m inside a thin-walled coaxial conducting cylindrical shell of radius R2=13.0R1 and the (same) length l. The net charge on the rod is Q1= +3.43 x 10^-12C; that on the shell is Q2= -2.32Q1. a) What is the magnitude and direction [radially inward or outward] of the electric field at radial distance r=2.00R2? b) What is the magnitude and direction at r=5.01R1? c) What is the charge on the interior and exterior surface of the shell? Give electric fields in N/C and charges in Coulombs.

2. A point particle with charge q=80uC is placed at a corner of a cube of edge a=8.1cm. What is the flux through each cube face forming the corner? and each of the other cube faces?

3. A small, nonconducting ball of mass m=1.3mg and charge q=2.6 x 10-8C (distributed uniformly through its volume) hangs from an insulating thread that makes an angle theta=39 degrees with a vertical, uniformly charged nonconducting sheet. Considering the gravitational force on the ball and assuming the sheet extends far vertically and into and out of the page, calculate the surface charge density of the sheet.

4. An electron enters a region of uniform electric field with an initial velocity of 48 km/s in the same direction as the electric field, which has magnitude E= 71 N/C. What is the speed in (km/s) of the electron 3.0 ns after entering this region? How far (in microns) does the electron travel during the 3.0 ns interval?

Solution Preview

Please refer to the attachment.

1. A conducting rod of radius R1 = 1.57 mm and length l = 13.9 m inside a thin-walled coaxial conducting cylindrical shell of radius R2 = 13.0R1 and the same length l. The net charge on the rod is Q1 = +3.43 x 10-12C; that on the shell is Q2= -2.32Q1. a) What is the magnitude E and direction [radially inward (type 1) or outward (type0)] of the electric field at radial distance r = 2.00R2? b) What is the magnitude E and the direction [(type 1) or outward (type0)] at r = 5.01R1? c) What is the charge on the interior and the exterior surface of the shell? Give electric fields in N/C and charges in Coulombs.

Solution:
Gaussian surface

Q2 = -2.32Q1= - 7.96x10-12C Q1 = +3.43 x 10-12C - - - - - - - - - - - - - - - - - - - - - - - - 40.82mm
+ + + + + + + + + + + + + + + + + R2=13R1=20.41mm
R1=1.57mm

+ + + + + + + + + + + + + + + + +
13.9 m

a) We will apply Gauss' law to determine the electric field E at r = 2R2 = 2 x 20.41 = 40.82 mm
Gauss' law: ∫E.ds = Q/ε0 ......(1) where ∫E.ds is the net flux passing through an imaginary closed surface (Gaussian surface) and Q is the net charge enclosed by the closed surface.
In this case, we consider as Gaussian surface a coaxial cylinder of radius 40.82 mm and length 13.9 m.
Let magnitude of electric field at a point on the Gaussian surface be E. Further, the direction of electric field vector is perpendicular to the Gaussian surface (straight surface of the cylinder).
We apply Gauss law over the Gaussian surface as follows:
Flux through each of the circular faces is zero as no field lines pass through the same.
Flux through the curved surface = E x Area of the curved surface = E x 2Пrl = E x 2П x 40.82 x 10-3 x 13.9 = 3.56E
Net flux through the Gaussian surface = 3.56E
Total ...

Solution Summary

The four problems on electric field, charge, flux, and charge density is provided. Step by step solutions provided.

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