Determine a formula for the change in surface area of a uniform solid sphere of radius r if its coefficient of linear expansion is alpha A (assume constant) and its temperature is changed by Delta T.Assume the temperature chan ge is not large, Delta T^2 ~ 0.

Area A = 4 pi R^2

lnA = ln(4 pi) + 2 ln R
dA/A = 2 dR/R = 2*(alpha)*dT
delta A = 2A*alpha*(delta T)

OR

A at T + deltaT
delta A = 4 pi (R + deltaR)^2 - A
= 4 pi R(1 + alpha*deltaT)^2 -A
= 4 pi R^2 *[1 + 2*alpha*deltaT +
(alpha*deltaT)^2] - A
= 2*A*alpha*deltaT + A*(alpha*deltaT)^2

(Ignore the second term in deltaT^2)

To determine the formula for the change in surface area of a uniform solid sphere, we know that the surface area of a sphere is given by the formula:

A = 4πr^2

Where A is the surface area and r is the radius of the sphere.

Given that the coefficient of linear expansion is α and the temperature change is ΔT, the change in the radius of the sphere can be expressed as:

Δr = αrΔT

This is derived from the equation for linear expansion, where the change in length is equal to the original length multiplied by the coefficient of linear expansion and the change in temperature.

Now, let's find the change in surface area (ΔA) due to the change in radius (Δr):

ΔA = 4π[(r + Δr)^2 - r^2]

Expanding the equation:

ΔA = 4π[(r^2 + 2rΔr + Δr^2) - r^2]

Simplifying:

ΔA = 4π[2rΔr + Δr^2]

Since ΔT^2 is negligible (ΔT^2 ~ 0), we can ignore the Δr^2 term:

ΔA ≈ 4π(2rΔr)

Substituting the value of Δr:

ΔA ≈ 4π(2r)(αrΔT)

Simplifying further:

ΔA ≈ 8παr^2ΔT

Therefore, the formula for the change in surface area of a uniform solid sphere with a given coefficient of linear expansion and temperature change is:

ΔA = 8παr^2ΔT

To determine the change in surface area of a uniform solid sphere of radius r when its temperature is changed by ΔT, we can use the formula for linear expansion and the formula for the surface area of a sphere.

The formula for linear expansion is given by:

ΔL = αL₀ΔT

Where ΔL is the change in length, α is the coefficient of linear expansion, L₀ is the initial length, and ΔT is the change in temperature.

In the case of a sphere, the change in length corresponds to the change in radius (Δr), and the initial length is the radius (r₀). So we can rewrite the formula as:

Δr = αr₀ΔT

To find the change in surface area, we need to consider how the surface area of a sphere changes with the change in radius. The formula for the surface area of a sphere is:

A = 4πr²

Differentiating both sides of the equation with respect to r, we get:

dA/dr = 8πr

This means that the change in surface area (ΔA) when the radius changes by Δr can be calculated using:

ΔA = 8πr₀Δr

Substituting the expression for Δr from the linear expansion formula, we have:

ΔA = 8πr₀(αr₀ΔT)

Simplifying:

ΔA = 8απr₀²ΔT

Therefore, the formula for the change in surface area of a uniform solid sphere of radius r, with a coefficient of linear expansion α and a temperature change ΔT, is:

ΔA = 8απr₀²ΔT