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*_{C}qQ/r^{2},

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*_{C}λ/d^{n},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/meter^{2}. 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*_{C}σ/d^{n},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

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