Suppose the drift velocity `v_(d)` in a material varied with the applied electric field E as `v_(d) prop sqrt(E)`. Then V – I graph for a wire made of such a material is best given by :

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the relationship between the drift velocity \( v_d \), the electric field \( E \), and the current \( I \) in the wire. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the relationship between drift velocity and electric field
We are given that the drift velocity \( v_d \) is proportional to the square root of the electric field \( E \):
\[ v_d \propto \sqrt{E} \]
This can be expressed mathematically as:
\[ v_d = k \sqrt{E} \]
where \( k \) is a proportionality constant.

### Step 2: Relate drift velocity to current
The current \( I \) in a conductor is given by the formula:
\[ I = n e A v_d \]
where:
- \( n \) is the number density of charge carriers,
- \( e \) is the charge of an electron,
- \( A \) is the cross-sectional area of the wire,
- \( v_d \) is the drift velocity.

### Step 3: Substitute the expression for drift velocity
Substituting the expression for \( v_d \) from Step 1 into the current equation:
\[ I = n e A (k \sqrt{E}) = n e A k \sqrt{E} \]
This shows that the current \( I \) is directly proportional to \( \sqrt{E} \):
\[ I \propto \sqrt{E} \]

### Step 4: Relate electric field to potential difference
The electric field \( E \) can also be expressed in terms of the potential difference \( V \) and the length \( L \) of the conductor:
\[ E = \frac{V}{L} \]

### Step 5: Substitute electric field in the current equation
Now, substituting \( E \) in terms of \( V \):
\[ I \propto \sqrt{\frac{V}{L}} \]
Since \( L \) is a constant for a given wire, we can simplify this to:
\[ I \propto \sqrt{V} \]

### Step 6: Express the relationship between voltage and current
From the relationship \( I \propto \sqrt{V} \), we can express it as:
\[ I = k' \sqrt{V} \]
where \( k' \) is a new constant.

### Step 7: Square both sides to find the relationship between \( V \) and \( I \)
Squaring both sides gives:
\[ I^2 = (k')^2 V \]
This indicates that the potential difference \( V \) is proportional to the square of the current \( I \):
\[ V \propto I^2 \]

### Step 8: Identify the shape of the V-I graph
The relationship \( V \propto I^2 \) describes a parabolic curve that opens upwards, which means the graph of \( V \) versus \( I \) is a parabola that is concave upwards.

### Conclusion
Thus, the V-I graph for a wire made of such a material is best represented by a parabolic curve concave upwards.

### Final Answer
The correct option is **B**.

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