`E` denotes electric field in a uniform conductor, `I` corresponding current through it, `v_(d)` velocity of electrons and `P` denotes thermal power produced in the conductor, then which of the following graph is incorrect?

Options

A.
B.
C.
D.

Explanation

To determine which graph is incorrect among the options provided, we will analyze the relationships between the electric field (E), current (I), drift velocity (v_d), and thermal power (P) produced in a uniform conductor step by step.

### Step 1: Analyze the relationship between drift velocity (v_d) and electric field (E)

The drift velocity (v_d) of electrons in a conductor can be expressed using the formula:

\[ v_d = \frac{eE\tau}{m} \]

Where:
- \( e \) is the charge of an electron,
- \( E \) is the electric field,
- \( \tau \) is the relaxation time,
- \( m \) is the mass of the electron.

From this equation, we can see that drift velocity (v_d) is directly proportional to the electric field (E):

\[ v_d \propto E \]

This means that if we plot v_d against E, we should get a straight line. Thus, the graph showing a linear relationship between v_d and E is correct.

### Step 2: Analyze the relationship between power (P) and electric field (E)

The thermal power (P) produced in a conductor can be expressed as:

\[ P = I^2 R \]

Where \( I \) is the current and \( R \) is the resistance. The current (I) can be expressed in terms of electric field (E) as:

\[ I = \frac{E A}{\rho} \]

Where:
- \( A \) is the cross-sectional area,
- \( \rho \) is the resistivity of the material.

Substituting this expression for current into the power equation gives:

\[ P = \left(\frac{E A}{\rho}\right)^2 R \]

This simplifies to:

\[ P \propto E^2 \]

This indicates that the relationship between power (P) and electric field (E) is quadratic, which means the graph should be a parabola. Thus, the graph showing a quadratic relationship between P and E is correct.

### Step 3: Analyze the relationship between power (P) and drift velocity (v_d)

From the previous steps, we know:

1. \( P \propto E^2 \)
2. \( v_d \propto E \)

If we square the relationship for drift velocity:

\[ E \propto v_d \implies E^2 \propto v_d^2 \]

Thus, substituting this into the power relationship gives:

\[ P \propto v_d^2 \]

This indicates that power (P) is also proportional to the square of drift velocity (v_d). Therefore, if we plot P against v_d, we should also get a quadratic relationship.

### Conclusion

From the analysis:
- The graph showing a linear relationship between v_d and E is correct.
- The graph showing a quadratic relationship between P and E is correct.
- The graph showing a quadratic relationship between P and v_d is also correct.

However, if any graph incorrectly represents these relationships, it would be the one that suggests a linear relationship between P and v_d, which is incorrect.

Thus, the incorrect graph is the one that suggests a linear relationship between power (P) and drift velocity (v_d).