Help with a Physics Question, Gaussian Surfaces.

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Hello guys. This is my first time here and your help would be greatly appreciated!

We have two thin concentric conducting spheres r1 and r2. THe spheres are connected with a power source which supplies emf of epsilon_0. The smaller sphere is connected to the positive pole of the power supply and the larger sphere is connected to the negative pole. Q is given. Using Gauss Law find the Electric Field between r1< R <r2 and find the electric field R>r2.

For this the electric field R. r2 would be 0 correct? Because charge enclosed is 0?

And for the electric field between r1 and r2 it would be equal to E = (+Q)/(4pi*R^2*Epsilon_0)?


After disconnecting the battery we ground the outer shell. Now calculate the electric field R>r2 and R<r1.

For R<r1, the electric field is 0 because charge enclosed is 0, correct?

And for R>r2 the electric field is now equal to E = (+Q)/(4pi*R^2*Epsilon_0) because charge enclosed is now +Q +Q -Q = +Q.

Could you guys please verify if my reasoning is correct for these problems. Thank you so much for your help!