Question 1  The potential due to a point charge Q at the origin may be written as V=Q4π?0r=Q4π?0x2+y2+z2√ Calculate Ex using equation Ex=−∂V∂x

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Question 1  The potential due to a point charge Q at the origin may be written as V=Q4π?0r=Q4π?0x2+y2+z2√

Calculate Ex using equation Ex=−∂V∂x.

Express your answer in terms of the given quantities and appropriate constants.

Part B

Calculate Ey using equation Ey=−∂V∂y.

Express your answer in terms of the given quantities and appropriate constants.

Part C

Calculate Ez using equation Ez=−∂V∂z.

Express your answer in terms of the given quantities and appropriate constants.

Part D

Show that the results of parts (a), (b), (c) agrees with the equation for the electric field of a point charge E? =14π?0qr2r^.

Question 2 .

A very long solid cylinder of radius R has positive charge uniformly distributed throughout it, with charge per unit volume ρ. From the expression for E for this cylinder: E(r)=λr2π?0R2 for r≤R and E(r)=λ2π?0r for r≥R, find the expressions for the electric potential V as a function of r, both inside and outside the cylinder. Let V = 0 at the surface of the cylinder. In each case, express your result in terms of the charge per unit length λ of the charge distribution.

Express your answer in terms of the given quantities and appropriate constants.

Part B

Express your answer in terms of the given quantities and appropriate constants.

Part C

Graph V as function of r from r = 0 to r = 3R.

Alternative Exercise 23.104

Part A

A very long solid cylinder of radius R has positive charge uniformly distributed throughout it, with charge per unit volume ρ. From the expression for E for this cylinder: E(r)=λr2π?0R2 for r≤R and E(r)=λ2π?0r for r≥R, find the expressions for the electric potential V as a function of r, both inside and outside the cylinder. Let V = 0 at the surface of the cylinder. In each case, express your result in terms of the charge per unit length λ of the charge distribution.

Express your answer in terms of the given quantities and appropriate constants.

V(r)(r>R) =

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Part B

Express your answer in terms of the given quantities and appropriate constants.

V(r)(r<R) =

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Part C

Graph V as function of r from r = 0 to r = 3R.

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Part D

Graph E as function of r from r = 0 to r = 3R.