Two charged conductors of unknown shape are situated in a vacuum. Conductor A ha

Question

Two charged conductors of unknown shape are situated in a vacuum. Conductor A has charge –Q0. Conductor B has charge +Q0. There is a direct path from conductor A to conductor B through the vacuum along the positive x-axis, beginning at x = xA and ending at x=xB (withxB >xA).

Suppose that the x-component of the electric field between xA and xB is given by
Ex = –DQ0x2., where D is a constant with appropriate units. Note that along the x-component of the electric field is negative.

A. What are the SI units of D? Express your answer in terms of N (newtons), C (coulombs), and m (meters). [Note: ‘C’ in this part means coulombs.]

B. Calculate the capacitance C of this conductor system. Your answer should be in terms of some or all of the given variables (D, Q0, xA, and/or xB) and physical constants. [Note: ‘C’ in this part means capacitance. You shouldn’t need to use the symbol ‘C’ that means the SI unit coulomb.]

Explanation / Answer

part a:

unit of Ex=N/C

unit of Q0=C

unit of x=m

so unit of D=unit of Ex/(unit of Q*unit of x^2)

=(N/C)/(C*m^2)=N/(C^2.m^2)

part B:

potential difference from x=xA to xB=integration of (-E.dx)

=integration of D*Q0*x^2*dx

=(D*Q0/3)*x^3

using the limits from x=xA to x=xB,we get potential difference=(D*Q0/3)*(xB^3-xA^3)

as for a capacitor, charge stored=capacitance*potential difference

==>capacitance=Q0/(D*Q0/3)*(xB^3-xA^3))

=3/(D*(xB^3-xA^3))

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