Rate of dissipation of Joule's heat in resistance per unit volume is (symbols have usual meaning)

Options

A. `sigmaE`
B. `sigmaJ`
C. `JE`
D. None of these

Answer

To find the rate of dissipation of Joule's heat in resistance per unit volume, we can follow these steps:

### Step 1: Understand Joule's Law
Joule's law states that the power (P) dissipated as heat in a resistor is given by the formula:
\[ P = V \cdot I \]
where \( V \) is the voltage across the resistor and \( I \) is the current flowing through it.

**Hint:** Remember that power is the rate at which energy is dissipated.

### Step 2: Relate Voltage and Current to Electric Field and Current Density
We can express voltage (V) in terms of electric field (E) and length (L) of the conductor:
\[ V = E \cdot L \]
Similarly, current (I) can be expressed in terms of current density (J) and cross-sectional area (A):
\[ I = J \cdot A \]

**Hint:** Think about how electric field and current density relate to the physical dimensions of the conductor.

### Step 3: Substitute the Expressions into the Power Formula
Substituting the expressions for voltage and current into the power formula gives:
\[ P = (E \cdot L) \cdot (J \cdot A) \]

**Hint:** Keep track of the units and dimensions as you substitute.

### Step 4: Simplify the Expression
Now, we can simplify this expression:
\[ P = E \cdot J \cdot (L \cdot A) \]
Since \( L \cdot A \) is the volume (V) of the conductor, we can rewrite the equation as:
\[ P = E \cdot J \cdot V \]

**Hint:** Remember that volume is the product of length and area.

### Step 5: Find the Rate of Dissipation of Joule's Heat per Unit Volume
To find the rate of dissipation of Joule's heat per unit volume, we divide the power by the volume:
\[ \text{Rate of dissipation per unit volume} = \frac{P}{V} = \frac{E \cdot J \cdot V}{V} = E \cdot J \]

**Hint:** Dividing by volume helps us find the rate per unit volume.

### Final Answer
Thus, the rate of dissipation of Joule's heat in resistance per unit volume is:
\[ E \cdot J \]