Dependence of intensity of gravitational field `(E)` of earth with distance `(r)` from centre of earth is correctly represented by

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Explanation

To solve the problem of how the intensity of the gravitational field \( E \) of the Earth depends on the distance \( r \) from the center of the Earth, we can follow these steps:

### Step 1: Understand Gravitational Field Intensity
The gravitational field intensity \( E \) at a distance \( r \) from the center of the Earth is defined as the force per unit mass experienced by a small test mass placed at that distance.

### Step 2: Use Newton's Law of Universal Gravitation
According to Newton's law of universal gravitation, the force \( F \) between two masses \( m_1 \) and \( m_2 \) is given by:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant.

### Step 3: Relate Gravitational Field to Force
The gravitational field intensity \( E \) can be expressed as:
\[ E = \frac{F}{m_2} \]
where \( m_2 \) is the mass of the test object. Substituting the expression for \( F \):
\[ E = \frac{G m_1}{r^2} \]
Here, \( m_1 \) is the mass of the Earth.

### Step 4: Analyze the Dependence on Distance
From the equation \( E = \frac{G m_1}{r^2} \), we can see that the gravitational field intensity \( E \) is inversely proportional to the square of the distance \( r \) from the center of the Earth. This means as \( r \) increases, \( E \) decreases.

### Step 5: Conclusion
Thus, the dependence of the intensity of the gravitational field \( E \) with distance \( r \) from the center of the Earth is correctly represented by the equation:
\[ E \propto \frac{1}{r^2} \]

### Final Answer
The correct representation of the dependence of the intensity of the gravitational field \( E \) of Earth with distance \( r \) from the center of Earth is \( E \propto \frac{1}{r^2} \).

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