Two Connected Charged Spheres Two spherical conductors of rad r_1 and r_2 are se

Question

Two Connected Charged Spheres Two spherical conductors of rad r_1 and r_2 are separated by a distance much greater than the radius of either sphere. The spheres are connected by a conducting wire as shown in the figure. The charges on the spheres in equilibrium are q_1 and q_2, respectively, and they are uniformly charged. Find the ratio of the magnitudes of the electric fields at the surfaces of the spheres. Conceptualize Imagine that the spheres are much farther apart than shown in the figure. Because they are so far apart, the field of one does not affect the charge distribution on the other. The conducting wire between them ensures that both spheres have the same electric potential. Categorize Because the spheres are so far apart, we model the charge distribution on them as spherically symmetric, and we can model the field and potential outside the spheres to be that due to point charges. Set the electric potential at the surfaces of the spheres equal to each other: V = k_e q_1/r_1 = k_e q_2/r_1 Solve for the ratio of charges on the spheres: q_1/q_2 = r_1/r_2 Write expressions for the magnitudes of the electric fields at the surfaces of the spheres: E_1 = k_e q_1/r_1^2 and E_2 = k_e q_2/r_2^2 Evaluate the ratio of these two fields: E_1/E_2 = q_1/q_2 r_2^2/r_1^2 Substitute for the ratio of charges from Equation (1): E_1/E_2 = r_1/r_2 r_2^2/r_1^2 (Use the following as necessary: r_1 and r_2.)

Explanation / Answer

The solution is fine till step 2

after step 2 , we need to further simplify that

E1/E2 = (r1/r2) (r2/r1)2

E1/E2 = r2/r1

ratio : r2/r1

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