Introduction:
In this problem, we have two concentric spherical metal shells of radii a and b (b>a). The inner shell has a charge q and the outer one is grounded. We need to determine the charge on the outer shell.

Explanation:
When the inner shell is charged with a charge q, the charge will distribute evenly over the surface of the inner shell. The charge on the inner shell will then induce an opposite charge on the outer shell, as the two shells are conductors and in contact with each other. Since the outer shell is grounded, the opposite charge will flow to the ground, leaving the outer shell with a net charge of -q.

The charge on the outer shell can be calculated using Gauss's law, which states that the electric flux through a closed surface is proportional to the charge enclosed within the surface. We can use a Gaussian surface in the form of a sphere with radius b, which encloses both the inner and outer shells. The electric flux through this surface is given by:

Φ = E * 4πb^2

Where E is the electric field at any point on the surface of the sphere. By applying Gauss's law, we know that:

Φ = q/ε₀

Where q is the charge enclosed within the Gaussian surface, and ε₀ is the permittivity of free space. Since the outer shell is grounded, the charge enclosed within the Gaussian surface is only the charge on the inner shell, which is q. Therefore:

E * 4πb^2 = q/ε₀

Solving for E, we get:

E = q/(4πε₀b^2)

This electric field is the same at every point on the surface of the Gaussian sphere. Therefore, the charge induced on the outer shell can be calculated by multiplying the electric field by the surface area of the outer shell. The surface area of the outer shell is given by:

A = 4πb^2 - 4πa^2

Therefore, the charge induced on the outer shell is:

q' = E * A

Substituting the value of E, we get:

q' = q/(4πε₀b^2) * (4πb^2 - 4πa^2)

Simplifying, we get:

q' = q * (b-a)/b

Therefore, the charge on the outer shell is -q * (b-a)/b.

Conclusion:
When the inner shell is charged with a charge q and the outer one is grounded, the charge induced on the outer shell is -q * (b-a)/b.