Using coulomb's law compute the electric field at p, a distance z above the center of a circular loop of radius r that carries a uniform line charge of density λ (see adja- cent figure). draw a diagram to explain how you arrived at the result. 2. find the electricotential a due to the same circular loop as in the previous problem. compute the gradient of the potential and verify that it is consistent with the field you found previously. hint: you may assume that the potential is zero at infinity and use the expression for potential due to an elemetal length along the loop and integrate.

The second largest public utility in the nation is the sole provider of electricity in 32 counties of southern florida. to meet the monthly demand for electricity in these counties, which is given by the inverse demand function p = 1,200 – 4q, the utility company has set up two electric generating facilities: q1 kilowatts are produced at facility 1, and q2 kilowatts are produced at facility 2 (so q = q1 + q2). the costs of producing electricity at each facility are given by c1(q1) = 8,000 + 6q12 and c2(q2) = 6,000 + 3q22, respectively. determine the profit-maximizing amounts of electricity to produce at the two facilities, the optimal price, and the utility company’s profit