Gravitational Field due to Spherical Shell of Matter

**A. Outside the shell:** \(E = -\frac{dv}{da} = -\frac{d}{da}\left(-\frac{GM}{a}\right) = GM\frac{d}{da}\left(\frac{1}{a}\right)\) \(E_{outside} = -G\frac{M}{a^2}\) **B. on the surface:** On the surface a = r. So, \(E_{surface}=-G\frac{M}{r^2}\) **C. Inside the shell:** \(E = -\frac{dv}{da} = -\frac{d}{da}\left(-\frac{GM}{ar}\right) = 0\) \(E_{inside} = 0\)

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