Ohm.s law in vector form is :

Options

A. V=I.R
B. `vecj= sigmavecE`
C. `vecE= sigma vecj`
D. `vecj= sigmavecE`

Explanation

To derive Ohm's law in vector form, we start with the fundamental relationships between voltage (V), current (I), resistance (R), and the physical quantities involved. Here are the steps to arrive at the vector form of Ohm's law:

### Step 1: Understand the basic relationship
Ohm's law states that the voltage (V) across a conductor is directly proportional to the current (I) flowing through it, given by the equation:
\[ V = I \cdot R \]
where R is the resistance.

### Step 2: Define current density
Current density (J) is defined as the current (I) flowing per unit area (A):
\[ J = \frac{I}{A} \]

### Step 3: Relate voltage to electric field
The electric field (E) can be defined in terms of voltage (V) and the length (L) over which the voltage is applied:
\[ E = \frac{V}{L} \]
Thus, we can express voltage as:
\[ V = E \cdot L \]

### Step 4: Substitute voltage in terms of electric field
Substituting the expression for V into the original Ohm's law equation gives:
\[ E \cdot L = I \cdot R \]

### Step 5: Relate resistance to resistivity
Resistance (R) can also be expressed in terms of resistivity (ρ), length (L), and area (A):
\[ R = \frac{\rho L}{A} \]
Substituting this into the equation gives:
\[ E \cdot L = I \cdot \frac{\rho L}{A} \]

### Step 6: Simplify the equation
Canceling L from both sides (assuming L is not zero) leads to:
\[ E = \frac{I \cdot \rho}{A} \]

### Step 7: Express current in terms of current density
From the definition of current density, we have:
\[ I = J \cdot A \]
Substituting this into the equation gives:
\[ E = \frac{J \cdot A \cdot \rho}{A} \]
This simplifies to:
\[ E = \rho J \]

### Step 8: Rearranging the equation
Rearranging this equation gives us:
\[ J = \frac{1}{\rho} E \]
Here, \(\frac{1}{\rho}\) is defined as conductivity (σ):
\[ J = \sigma E \]

### Conclusion
Thus, the vector form of Ohm's law is:
\[ \mathbf{J} = \sigma \mathbf{E} \]
where \(\mathbf{J}\) is the current density vector, \(\sigma\) is the conductivity, and \(\mathbf{E}\) is the electric field vector.