Derivation of Ohm's Law from Current Density

Ohm’s Law can be derived from the basic relation between current density and electric field.

Step 1: Define Current Density

Current density \( J \) is defined as the current \( I \) per unit area \( A \):

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

Step 2: Relate Current Density to Drift Velocity

The current density is also given by:

\[ J = n e v_d \]

where:

  • \( n \) = Number of charge carriers per unit volume
  • \( e \) = Charge on each carrier
  • \( v_d \) = Drift velocity

Step 3: Drift Velocity and Electric Field

Drift velocity \( v_d \) is proportional to the applied electric field \( E \):

\[ v_d = \mu E \]

where \( \mu \) is the mobility of charge carriers.

Step 4: Substitute into Current Density

Substitute \( v_d = \mu E \) into \( J = n e v_d \):

\[ J = n e \mu E \]

Step 5: Introduce Conductivity

Let \( \sigma = n e \mu \) be the electrical conductivity of the material. Then:

\[ J = \sigma E \]

Step 6: Rearranging to Get Ohm’s Law

From the above, the electric field is:

\[ E = \frac{J}{\sigma} \]

But the electric field across a conductor of length \( L \) and voltage \( V \) is:

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

Step 7: Combine to Express Current

So, \[ J = \sigma \frac{V}{L} \] and since \( J = \frac{I}{A} \), we get: \[ \frac{I}{A} = \sigma \frac{V}{L} \]

Final Step: Express Ohm's Law

Rearranging: \[ I = \left( \sigma \frac{A}{L} \right) V \] Let \( R = \frac{L}{\sigma A} \) be the resistance of the conductor. Then: \[ V = IR \]

Conclusion:

This is the mathematical statement of Ohm’s Law, derived using current density and conductivity concepts: \[ V = I R \]

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