Final answer:
The electrical power dissipated per unit volume in a resistor is found by deriving the power formula P = I²R to include volume and is given by p = J²ρ, where J is the current density and ρ is the resistivity.
Step-by-step explanation:
The question asks about the electrical power dissipated per unit volume, denoted as p, in a resistor with uniform cross-sectional area A, length L, and uniform resistivity ρ through which a current with uniform current density J is flowing. Using the formula P = I²R, we can derive the expression for p by relating resistance R to the resistivity ρ, area A, and length L.
First, we find the resistance of the resistor R = ρL / A. Then, since the current density J is the current per unit area, we express current as I = JA. Substituting I and R into the power formula, we get P = J²A²×(ρL / A) = J²ρLA. To find power per unit volume, we divide by the volume V = AL, thus p = P / V = J²ρ. Hence, the power dissipated per unit volume p is proportional to the square of the current density J and the resistivity ρ.