Electric field (E) and current density `(J)` have relation

Options

A. `Eprop J`
B. `E propJ^(2)`
C. `E prop(1)/(J^(2))`
D. `E^(2)prop(1)/(J)`

Answer

To find the relationship between electric field (E) and current density (J), we can follow these steps:

### Step 1: Understand the Definitions
- **Electric Field (E)**: It is defined as the potential difference (V) across a conductor divided by the length (L) of the conductor.

\[ E = \frac{V}{L} \]

- **Current Density (J)**: It is defined as the current (I) flowing through a conductor per unit cross-sectional area (A).

\[ J = \frac{I}{A} \]

### Step 2: Apply Ohm's Law
According to Ohm's law, the current (I) flowing through a conductor can be expressed as:

\[ I = \frac{V}{R} \]

Where R is the resistance of the conductor. The resistance can be expressed in terms of resistivity (ρ), length (L), and area (A):

\[ R = \frac{\rho L}{A} \]

### Step 3: Substitute Resistance in Ohm's Law
Substituting the expression for R into Ohm's law gives:

\[ I = \frac{V}{\frac{\rho L}{A}} = \frac{V \cdot A}{\rho L} \]

### Step 4: Relate Current Density to Electric Field
Now, we can express the current density (J) in terms of the electric field (E):

1. From the definition of current density:
\[ J = \frac{I}{A} \]
Substitute I from the previous step:
\[ J = \frac{V \cdot A}{\rho L \cdot A} = \frac{V}{\rho L} \]

2. Now, substitute the expression for E:
\[ J = \frac{E \cdot L}{\rho L} \]

3. The length (L) cancels out:
\[ J = \frac{E}{\rho} \]

### Step 5: Establish the Relationship
From the equation \( J = \frac{E}{\rho} \), we can see that:

- Current density (J) is directly proportional to the electric field (E) when resistivity (ρ) is constant.

### Final Result
Thus, the relationship between electric field (E) and current density (J) is given by:

\[ J \propto E \]