Calculating the Changes

When the temperature increases by \( T \), the volume change of the liquid can be expressed using the coefficient of volume expansion:

Change in volume of the liquid, \( \Delta V = V_0 \cdot \beta \cdot T \)

Where \( V_0 \) is the initial volume of the liquid. The increase in volume will exert pressure on the walls of the shell. The relationship between pressure change and volume change is given by the bulk modulus:

Using the definition of bulk modulus:

\( K = -\frac{dP}{\frac{\Delta V}{V_0}} \)

Rearranging this gives us:

\( dP = -K \cdot \frac{\Delta V}{V_0} \)

Substituting the expression for \( \Delta V \):

\( dP = -K \cdot \frac{V_0 \cdot \beta \cdot T}{V_0} = -K \cdot \beta \cdot T \)

Considering the Shell Expansion

Now, we also need to account for the expansion of the copper shell. The volume expansion of the shell can be related to its linear expansion:

For a spherical shell, the volume change due to linear expansion is approximately:

\( \Delta V_{shell} \approx 3V_0 \cdot \alpha \cdot T \)

Thus, the pressure exerted by the expansion of the shell can be expressed similarly:

\( dP_{shell} = K_{shell} \cdot \frac{\Delta V_{shell}}{V_{shell}} \)

However, since we are primarily interested in the outward pressure due to the liquid, we can combine the effects of both the liquid and the shell. The total outward pressure will be influenced by both the liquid's bulk modulus and the shell's expansion.

Final Expression

Combining these effects, we find that the net outward pressure \( dP \) can be expressed as:

\( dP = K \cdot \beta \cdot T + 3K \cdot \alpha \cdot T \)

To simplify, we can factor out \( T \):

\( dP = T \cdot (K \cdot \beta + 3K \cdot \alpha) \)

Now, substituting the values of \( \beta \) and \( \alpha \) into the equation, we can analyze the options provided in the question. The correct answer will depend on the specific values of \( K \), \( \beta \), and \( \alpha \). However, based on the relationships derived, we can conclude that the outward pressure \( dP \) is influenced by the bulk modulus and the coefficients of expansion.

Conclusion

In summary, the outward pressure \( dP \) on the copper shell due to the increase in temperature can be expressed in terms of the bulk modulus of the liquid and the coefficients of expansion of both the liquid and the shell. The final expression will depend on the specific values of these properties, but the approach outlined here provides a clear pathway to derive the relationship.