8.1.P5

The functional dependence of the electric field from some highly symmetric shapes can be determined rather straightforwardly from dimensional analysis. Given that the electric field is defined by the equation F = qE and the force on a point charge q from a charge Q a distance r away is F = k_CqQ/r², 

A. Find the dimensionality of the Coulomb constant k_C and the electric field, E, in terms of the basic dimensions, M, L, T, and Q. Show your work.

B. For a long uniform line of charge, we know neither the total charge on the line nor its length, only the charge per unit length, λ, with units of Coulomb/meter. Assuming that the dependence of the E-field produced by that line as a function of the (perpendicular) distance d from the line will look something like E = αk_Cλ/dⁿ,where α is a dimensionless constant. Use dimensional analysis to find the value of n. If you can find it, show your work below. If you can't, explain why not.

C. For a large flat uniform sheet of charge, we know neither the total charge on the sheet nor its area, only the charge per unit area, σ, with units of Coulomb/meter². Assuming that the dependence of the E-field produced by that line as a function of the (perpendicular) distance d from the sheet will look something like E = αk_Cσ/dⁿ,where α is a dimensionless constant. Use dimensional analysis to find the value of n. If you can find it, show your work below. If you can't, explain why not.

Joe Redish 3/4/16