If V and u are electric potential and energy density, respectively, at a distance r from a positive point charge, then which of the following graph is correct ?

Options

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B.
C.
D.

Answer

To solve the problem, we need to establish the relationship between electric potential (V) and energy density (U) at a distance r from a positive point charge. Here’s a step-by-step solution:

### Step 1: Understand the Electric Potential (V)
The electric potential (V) at a distance r from a point charge Q is given by the formula:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant.

### Step 2: Understand the Electric Field (E)
The electric field (E) at a distance r from a point charge Q is given by:
\[ E = \frac{kQ}{r^2} \]

### Step 3: Calculate the Energy Density (U)
The energy density (U) in an electric field is given by the formula:
\[ U = \frac{1}{2} \epsilon_0 E^2 \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 4: Substitute E into the Energy Density Formula
Now, substituting the expression for E into the energy density formula:
\[ U = \frac{1}{2} \epsilon_0 \left( \frac{kQ}{r^2} \right)^2 \]
\[ U = \frac{1}{2} \epsilon_0 \frac{k^2 Q^2}{r^4} \]

### Step 5: Relate U to V
From the expression for V, we can express r in terms of V:
\[ r = \frac{kQ}{V} \]
Now substituting this into the expression for U:
\[ U = \frac{1}{2} \epsilon_0 \frac{k^2 Q^2}{\left( \frac{kQ}{V} \right)^4} \]
\[ U = \frac{1}{2} \epsilon_0 \frac{k^2 Q^2 V^4}{k^4 Q^4} \]
\[ U = \frac{\epsilon_0 V^4}{2 k^2 Q^2} \]

### Step 6: Establish the Relationship
From the above expression, we can see that:
\[ U \propto V^4 \]
This means that energy density (U) is proportional to the fourth power of electric potential (V).

### Step 7: Graphical Representation
If we plot U on the x-axis and V on the y-axis, the relationship \( U \propto V^4 \) suggests that the graph will be a parabolic curve opening upwards.

### Step 8: Determine the Correct Graph
Since U is plotted on the x-axis and V on the y-axis, the relationship can be expressed as:
\[ U = k V^4 \]
This indicates that the graph will be a curve that passes through the origin and rises steeply as V increases.

### Conclusion
The correct graph representing the relationship between electric potential (V) and energy density (U) is a parabolic curve.

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