In Newton's law of cooling `(d theta)/(dt)=-k(theta-theta_(0))` the constant `k` is proportional to .

Options

A. `A`, surface area of the body
B. `S`, specific heat of the body
C. `(1)/(m)`, m being mass of the body
D. e, emissity of the body

Explanation

To determine what the constant \( k \) in Newton's law of cooling is proportional to, we need to analyze the factors that influence the rate of heat transfer between an object and its surroundings.

### Step 1: Understanding Newton's Law of Cooling
Newton's law of cooling states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings. Mathematically, this is expressed as:

\[ \frac{d\theta}{dt} = -k(\theta - \theta_0) \]

where:
- \( \theta \) is the temperature of the object,
- \( \theta_0 \) is the ambient temperature,
- \( k \) is the cooling constant.

### Step 2: Identifying Factors Affecting \( k \)
The cooling constant \( k \) is influenced by several physical factors:

1. **Surface Area**: The larger the surface area of the object, the more heat can be transferred to or from the surroundings. Therefore, \( k \) is proportional to the surface area of the object.

2. **Specific Heat**: The specific heat of the material affects how quickly it can change temperature. Materials with lower specific heat will change temperature more rapidly, influencing the value of \( k \).

3. **Emissivity**: This is a measure of how effectively a surface emits thermal radiation. A higher emissivity means that the surface can radiate heat more efficiently, which also affects the value of \( k \).

### Step 3: Conclusion
From the analysis, we conclude that the constant \( k \) is proportional to the following factors:
- Surface area
- Specific heat
- Emissivity

Thus, we can summarize that:

\[ k \propto \text{Surface Area, Specific Heat, Emissivity} \]