Askiitians Tutor Team
Last Activity: 5 Months ago
To determine the thermal energy produced in a resistor when alternating currents pass through it, we can use the relationship between current, resistance, and power. The thermal energy generated in a resistor is directly related to the power dissipated over time. Let's break this down step by step for both currents and then analyze the situation when they are combined.
Thermal Energy from Current I1
The first current is given by I1 = I0 sin(ωt). The power dissipated in a resistor can be calculated using the formula:
Substituting our expression for I1, we have:
- P1 = (I0 sin(ωt))² R = I0² sin²(ωt) R
To find the average power over one complete cycle (time period T), we need to integrate this expression over one period and then divide by T:
- Average Power (P_avg) = (1/T) ∫ P1 dt
Since the average value of sin²(ωt) over one complete cycle is 1/2, we can simplify:
- P_avg = (1/T) ∫ (I0² sin²(ωt) R) dt = (I0² R / 2)
The thermal energy produced in one time period (T) is then:
- Energy (E1) = P_avg × T = (I0² R / 2) × T = (I0² R T / 2)
Thermal Energy from Current I2
The second current is I2 = -I0 sin(ωt). The negative sign indicates that the current is flowing in the opposite direction, but the power dissipated in the resistor remains the same because power depends on the square of the current:
- P2 = (-I0 sin(ωt))² R = I0² sin²(ωt) R
Following the same steps as before, we find:
Thus, the thermal energy produced in one time period for I2 is:
- Energy (E2) = (I0² R T / 2)
Combined Currents I1 and I2
When both currents flow through the resistor simultaneously, we can express the total current as:
- I_total = I1 + I2 = I0 sin(ωt) - I0 sin(ωt) = 0
Since the total current is zero at all times, the power dissipated in the resistor is:
- P_total = (I_total)² R = 0
This means that no thermal energy is produced when both currents are present simultaneously:
Principle of Superposition
The principle of superposition states that in a linear system, the total response (in this case, thermal energy) caused by multiple stimuli (currents) is equal to the sum of the responses that would have been caused by each stimulus individually. Here, we see that:
- E1 + E2 = (I0² R T / 2) + (I0² R T / 2) = I0² R T
- E_total = 0
Since the total energy produced when both currents are combined is zero, the principle of superposition does not hold in this case. The opposing currents completely cancel each other out, leading to no thermal energy being generated.
In summary, while each current produces thermal energy individually, their simultaneous presence leads to a cancellation effect, demonstrating an interesting aspect of alternating current behavior in resistive circuits.
