To determine the thermal energy produced in a resistor by alternating currents, we can use the formula for power dissipated in a resistor and integrate it over one complete cycle. Let's break this down step by step for both currents, and then analyze the situation when both currents flow simultaneously.
Thermal Energy from Current i1
The first current is given by i1 = i0 sin(wt). The power dissipated in a resistor due to this current can be expressed as:
P = i^2 R
Substituting for i1, we have:
P1 = (i0 sin(wt))^2 R = i0^2 R sin^2(wt)
To find the thermal energy produced over one complete time period (T), we need to integrate the power over that period:
Energy (E1) = ∫ P1 dt from 0 to T
Since the average value of sin²(wt) over one complete cycle is 1/2, we can simplify the integral:
E1 = ∫ (i0^2 R sin²(wt)) dt = i0^2 R (T/2)
Thus, the thermal energy produced by current i1 in one time period is:
E1 = (i0^2 R T) / 2
Thermal Energy from Current i2
Now, let's analyze the second current, which is i2 = -i0 sin(wt). The power dissipated in the resistor is:
P2 = (-i0 sin(wt))^2 R = i0^2 R sin^2(wt)
Notice that the expression for power is the same as for i1, since squaring the current eliminates the negative sign:
E2 = (i0^2 R T) / 2
Simultaneous Currents i1 and i2
When both currents flow through the resistor simultaneously, we can use the principle of superposition. The total current, i_total = i1 + i2 = i0 sin(wt) - i0 sin(wt) = 0. This means that at any point in time, the currents cancel each other out completely.
As a result, the power dissipated in the resistor when both currents are present is:
P_total = (0)^2 R = 0
Therefore, the thermal energy produced when both currents are present is:
E_total = 0
Principle of Superposition
The principle of superposition states that in a linear system, the net response at a given time or space caused by multiple stimuli is equal to the sum of the responses that would have been caused by each stimulus individually. In this case, since the two currents are equal in magnitude but opposite in direction, they perfectly cancel each other out, leading to zero thermal energy production.
In summary, the thermal energy produced by each current individually is (i0^2 R T) / 2, while the combined effect of both currents results in zero thermal energy due to their cancellation. This scenario indeed obeys the principle of superposition, as the total effect is the sum of the individual effects, which in this case results in no energy being dissipated in the resistor.
