Use fundamental units to show that equation P=IV is homogenous, where I is electric current , V is voltage and P is power.
The Correct Answer and Explanation is:
To demonstrate that the equation ( P = IV ) is homogeneous, we need to express power (( P )), current (( I )), and voltage (( V )) in terms of their fundamental SI units. This will show that both sides of the equation have the same dimensions.
- Defining the Variables:
- Power (P): In the SI system, power is measured in watts (W). One watt is defined as one joule per second:
[
P \text{ (in watts)} = 1 \text{ W} = 1 \frac{\text{J}}{\text{s}}
] - Current (I): Electric current is measured in amperes (A). The definition of one ampere is the flow of one coulomb of charge per second:
[
I \text{ (in amperes)} = 1 \text{ A} = 1 \frac{\text{C}}{\text{s}}
] - Voltage (V): Voltage is measured in volts (V). One volt is defined as one joule per coulomb:
[
V \text{ (in volts)} = 1 \text{ V} = 1 \frac{\text{J}}{\text{C}}
]
- Substituting into the Equation:
The equation ( P = IV ) can be expressed using these definitions:
[
P = I \cdot V
]
Substituting the units:
[
P = \left(1 \frac{\text{C}}{\text{s}}\right) \cdot \left(1 \frac{\text{J}}{\text{C}}\right)
] - Simplifying the Units:
The units on the right-hand side can be simplified:
[
P = 1 \frac{\text{C} \cdot \text{J}}{\text{s} \cdot \text{C}} = 1 \frac{\text{J}}{\text{s}} = 1 \text{ W}
]
Since we have shown that the left side ( P ) has the same units as the right side ( IV ) (both are measured in watts), we can conclude that the equation ( P = IV ) is homogeneous. This means that the dimensions on both sides of the equation are consistent and confirm that the physical relationship between power, current, and voltage is valid in terms of their units. Homogeneity in an equation is crucial as it ensures the physical validity and consistency of the mathematical model being used.