A thermally isulated piece of metal is heated under atmospheric pressure by an electric current so that it receives electric energy at a constant power P. This leads to an increase of absolute temperature T of the metal with time t as follows:
`T(t)=T_0[1+a(t-t_0)]^(1//4)`. Here, a, `t_0` and `T_0` are constants. The heat capacity `C_p(T)` of the metal is

Options

A. `(4P)/(aT_0)`
B. `(4PT)/(aT_0^4)`
C. `(2PT)/(aT_0^4)`
D. `(2P)/(aT_0)`

Explanation

To solve the problem, we need to find the heat capacity \( C_p(T) \) of the metal as a function of temperature \( T \). We start from the relationship given in the question:

\[ T(t) = T_0 [1 + a(t - t_0)]^{1/4} \]

### Step 1: Understand the definition of heat capacity
The heat capacity \( C_p \) is defined as the amount of heat \( DQ \) added to the system per unit increase in temperature \( D T \):

\[ C_p = \frac{DQ}{DT} \]

### Step 2: Relate heat \( DQ \) to power \( P \)
Since the metal is heated by an electric current at constant power \( P \), we can express the heat added \( DQ \) in terms of power and time:

\[ DQ = P \cdot dt \]

### Step 3: Differentiate the temperature function
To find \( C_p \), we need to differentiate the temperature function \( T(t) \) with respect to time \( t \). First, we rewrite the temperature equation:

\[ T = T_0 [1 + a(t - t_0)]^{1/4} \]

Now, we differentiate \( T \) with respect to \( t \):

\[ \frac{dT}{dt} = \frac{1}{4} T_0 [1 + a(t - t_0)]^{-3/4} \cdot a \]

### Step 4: Substitute \( \frac{dT}{dt} \) into the heat capacity formula
Now, substituting \( DQ \) and \( \frac{dT}{dt} \) into the heat capacity formula:

\[ C_p = \frac{DQ}{DT} = \frac{P \cdot dt}{\frac{dT}{dt} \cdot dt} = \frac{P}{\frac{dT}{dt}} \]

### Step 5: Substitute \( \frac{dT}{dt} \) into the equation
From the previous differentiation, we have:

\[ C_p = \frac{P}{\frac{1}{4} T_0 [1 + a(t - t_0)]^{-3/4} \cdot a} \]

### Step 6: Simplify the expression
Now, simplify the expression for \( C_p \):

\[ C_p = \frac{4P [1 + a(t - t_0)]^{3/4}}{a T_0} \]

### Step 7: Express \( C_p \) in terms of \( T \)
Recall that \( T = T_0 [1 + a(t - t_0)]^{1/4} \). Therefore, we can express \( [1 + a(t - t_0)] \) in terms of \( T \):

\[ [1 + a(t - t_0)] = \left(\frac{T}{T_0}\right)^4 \]

Substituting this back into the equation for \( C_p \):

\[ C_p = \frac{4P \left(\frac{T}{T_0}\right)^3}{a T_0} \]

### Final Expression for Heat Capacity
Thus, the heat capacity as a function of temperature \( T \) is:

\[ C_p(T) = \frac{4P}{a T_0^4} T^3 \]