Question:
An iron sphere having diameter \(D\) and mass \(M\) is immersed in hot water so that the temperature of the sphere increases by \(\delta T\). If \(\alpha\) is the coefficient of linear expansion of the iron then the change in the surface area of the sphere is
An iron sphere having diameter \(D\) and mass \(M\) is immersed in hot water so that the temperature of the sphere increases by \(\delta T\). If \(\alpha\) is the coefficient of linear expansion of the iron then the change in the surface area of the sphere is
Show Hint
For thermal expansion,
\[
L'=L(1+\alpha\Delta T)
\]
and area expansion can be obtained by squaring the linear dimension.
Updated On: Jun 22, 2026
- \(\pi D^2\alpha\delta T(\alpha\delta T-4)\)
- \(\pi D^2\alpha\delta T(\alpha\delta T+4)\)
- \(\pi D^2\alpha\delta T(\alpha\delta T-2)\)
- \(\pi D^2\alpha\delta T(\alpha\delta T+2)\)
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The Correct Option is D
Solution and Explanation
Step 1: Write the initial surface area of the sphere.
Surface area of a sphere is \[ A=\pi D^2 \] where \(D\) is the diameter of the sphere.
Step 2: Find the new diameter after expansion.
After heating through temperature rise \(\delta T\), \[ D'=D(1+\alpha\delta T) \]
Step 3: Find the new surface area.
New surface area is \[ A'=\pi (D')^2 \] Substituting \(D'\), \[ A'=\pi D^2(1+\alpha\delta T)^2 \] Expanding, \[ A'=\pi D^2\left(1+2\alpha\delta T+\alpha^2\delta T^2\right) \]
Step 4: Calculate the change in surface area.
\[ \Delta A=A'-A \] \[ \Delta A = \pi D^2\left(1+2\alpha\delta T+\alpha^2\delta T^2\right)-\pi D^2 \] \[ \Delta A = \pi D^2\left(2\alpha\delta T+\alpha^2\delta T^2\right) \] Taking common factor \(\alpha\delta T\), \[ \Delta A = \pi D^2\alpha\delta T(\alpha\delta T+2) \]
Step 5: Final conclusion.
Hence, the change in surface area is \[ \boxed{ \pi D^2\alpha\delta T(\alpha\delta T+2) } \]
Surface area of a sphere is \[ A=\pi D^2 \] where \(D\) is the diameter of the sphere.
Step 2: Find the new diameter after expansion.
After heating through temperature rise \(\delta T\), \[ D'=D(1+\alpha\delta T) \]
Step 3: Find the new surface area.
New surface area is \[ A'=\pi (D')^2 \] Substituting \(D'\), \[ A'=\pi D^2(1+\alpha\delta T)^2 \] Expanding, \[ A'=\pi D^2\left(1+2\alpha\delta T+\alpha^2\delta T^2\right) \]
Step 4: Calculate the change in surface area.
\[ \Delta A=A'-A \] \[ \Delta A = \pi D^2\left(1+2\alpha\delta T+\alpha^2\delta T^2\right)-\pi D^2 \] \[ \Delta A = \pi D^2\left(2\alpha\delta T+\alpha^2\delta T^2\right) \] Taking common factor \(\alpha\delta T\), \[ \Delta A = \pi D^2\alpha\delta T(\alpha\delta T+2) \]
Step 5: Final conclusion.
Hence, the change in surface area is \[ \boxed{ \pi D^2\alpha\delta T(\alpha\delta T+2) } \]
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