# Exercise 22.2:

Exercise 22.2:

The flux is given by:

where E is the magnitude of the electric field, A is the area of the plate, and θ is the angle between the field and a line perpendicular to the surface of the plate.

In this case, θ = 90º – 20º = 70º

The flux is then:

A point charge q1 = 3.75 nC is located on the x-axis at x = 1.95 m , and a second point charge q2 = -5.80 nC is on the y-axis at y = 1.10 m .

A) What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with radius r1 = 0.545 m ?

B) What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with radius r2 = 1.45 m ?

C) What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with radius r3 = 2.80 m ?

14 ) A solid metal sphere with radius 0.490 m carries a net charge of 0.300 nC .

A) Find the magnitude of the electric field at a point 0.119 m outside the surface of the sphere. Express your answer using three significant figures.

18) The electric field 0.300 m from a very long uniform line of charge is 710 N/C .

How much charge is contained in a section of the line of length 1.40 cm ?

20) At a distance of 0.194 cm from the center of a charged conducting sphere with radius 0.100cm, the electric field is 465 N/C . What is the electric field 0.592 cm from the center of the sphere?

B) At a distance of 0.200 cm from the axis of a very long charged conducting cylinder with radius 0.100cm, the electric field is 465 N/C . What is the electric field 0.614 cm from the axis of the cylinder?

C) At a distance of 0.184 cm from a large uniform sheet of charge, the electric field is 465 N/C . What is the electric field 1.26 cm from the sheet?

24) Charge Q is distributed uniformly throughout the volume of an insulating sphere of radius R = 4.00 cm. At a distance of r = 8.00 cm from the center of the sphere, the electric field due to the charge distribution has magnitude 890 N/C .

A) What is the volume charge density for the sphere?

B) What is the magnitude of the electric field at a distance of 2.00 cm from the sphere's center?

28) A square insulating sheet 90.0 cm on a side is held horizontally. The sheet has 6.50 nC of charge spread uniformly over its area.

A) Calculate the magnitude of the electric field at a point 0.100 mm above the center of the sheet.

B) Estimate the magnitude of the electric field at a point located a distance 100 m above the center of the sheet.

E = 1.67×10−4 N/C

E = 7.31×10−4 N/C

E = 5.84×10−3 N/C

E = 8.18×10−2 N/C

C) Would the answers to parts A and B be different if the sheet were made of a conducting material? Select the correct answer and explanation.

30) Two very large, non-conducting plastic sheets, each 10.0 cm thick, carry uniform charge densities σ1, σ2, σ3 and σ4 on their surfaces, as shown in the following figure (Figure 1) . These surface charge densities have the values σ1 = -7.30 μC/m2 , σ2 = 5.00 μC/m2, σ3 = 3.00 μC/m2 , and σ4 = 4.00 μC/m2. Use Gauss's law to find the magnitude and direction of the electric field at the following points, far from the edges of these sheets.

A) What is the magnitude of the electric field at point A, 5.00 cm from the left face of the left-hand sheet?

B) What is the magnitude of the electric field at point B, 1.25 cm from the inner surface of the right-hand sheet?

C) What is the magnitude of the electric field at point C, in the middle of the right-hand sheet?

42) A solid conducting sphere carrying charge q has radius a. It is inside a concentric hollow conducting sphere with inner radius b and outer radius c. The hollow sphere has no net charge.

A) Derive an expression for the electric-field magnitude in terms of the distance r from the center for the region r<a.

B) Derive an expression for the electric-field magnitude in terms of the distance r from the center for the region a<r<b.

C) Derive an expression for the electric-field magnitude in terms of the distance r from the center for the region b<r<c.

D) Derive an expression for the electric field magnitude in terms of the distance r from the center for the region rc.

E) What is the charge on the inner surface of the hollow sphere?

The flux is given by:

where E is the magnitude of the electric field, A is the area of the plate, and θ is the angle between the field and a line perpendicular to the surface of the plate.

In this case, θ = 90º – 20º = 70º

The flux is then:

A point charge q1 = 3.75 nC is located on the x-axis at x = 1.95 m , and a second point charge q2 = -5.80 nC is on the y-axis at y = 1.10 m .

A) What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with radius r1 = 0.545 m ?

B) What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with radius r2 = 1.45 m ?

C) What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with radius r3 = 2.80 m ?

14 ) A solid metal sphere with radius 0.490 m carries a net charge of 0.300 nC .

A) Find the magnitude of the electric field at a point 0.119 m outside the surface of the sphere. Express your answer using three significant figures.

18) The electric field 0.300 m from a very long uniform line of charge is 710 N/C .

How much charge is contained in a section of the line of length 1.40 cm ?

20) At a distance of 0.194 cm from the center of a charged conducting sphere with radius 0.100cm, the electric field is 465 N/C . What is the electric field 0.592 cm from the center of the sphere?

B) At a distance of 0.200 cm from the axis of a very long charged conducting cylinder with radius 0.100cm, the electric field is 465 N/C . What is the electric field 0.614 cm from the axis of the cylinder?

C) At a distance of 0.184 cm from a large uniform sheet of charge, the electric field is 465 N/C . What is the electric field 1.26 cm from the sheet?

24) Charge Q is distributed uniformly throughout the volume of an insulating sphere of radius R = 4.00 cm. At a distance of r = 8.00 cm from the center of the sphere, the electric field due to the charge distribution has magnitude 890 N/C .

A) What is the volume charge density for the sphere?

B) What is the magnitude of the electric field at a distance of 2.00 cm from the sphere's center?

28) A square insulating sheet 90.0 cm on a side is held horizontally. The sheet has 6.50 nC of charge spread uniformly over its area.

A) Calculate the magnitude of the electric field at a point 0.100 mm above the center of the sheet.

B) Estimate the magnitude of the electric field at a point located a distance 100 m above the center of the sheet.

E = 1.67×10−4 N/C

E = 7.31×10−4 N/C

E = 5.84×10−3 N/C

E = 8.18×10−2 N/C

C) Would the answers to parts A and B be different if the sheet were made of a conducting material? Select the correct answer and explanation.

30) Two very large, non-conducting plastic sheets, each 10.0 cm thick, carry uniform charge densities σ1, σ2, σ3 and σ4 on their surfaces, as shown in the following figure (Figure 1) . These surface charge densities have the values σ1 = -7.30 μC/m2 , σ2 = 5.00 μC/m2, σ3 = 3.00 μC/m2 , and σ4 = 4.00 μC/m2. Use Gauss's law to find the magnitude and direction of the electric field at the following points, far from the edges of these sheets.

A) What is the magnitude of the electric field at point A, 5.00 cm from the left face of the left-hand sheet?

B) What is the magnitude of the electric field at point B, 1.25 cm from the inner surface of the right-hand sheet?

C) What is the magnitude of the electric field at point C, in the middle of the right-hand sheet?

42) A solid conducting sphere carrying charge q has radius a. It is inside a concentric hollow conducting sphere with inner radius b and outer radius c. The hollow sphere has no net charge.

A) Derive an expression for the electric-field magnitude in terms of the distance r from the center for the region r<a.

B) Derive an expression for the electric-field magnitude in terms of the distance r from the center for the region a<r<b.

C) Derive an expression for the electric-field magnitude in terms of the distance r from the center for the region b<r<c.

D) Derive an expression for the electric field magnitude in terms of the distance r from the center for the region rc.

E) What is the charge on the inner surface of the hollow sphere?

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