In a sine wave, the relationship between current and voltage amplitudes is determined by the impedance of the circuit through which the sine wave is passing. Impedance is a measure of opposition to the flow of current in an alternating current (AC) circuit. It consists of two components: resistance (measured in ohms) and reactance (also measured in ohms).
For simplicity, let's consider a purely resistive circuit, which means there is no reactive component like capacitance or inductance. In such a circuit, the voltage-current relationship for sine waves is governed by Ohm's law:
Voltage (V) = Current (I) * Resistance (R)
From this equation, we can derive the relationship between the voltage and current amplitudes for two sine waves with equal frequencies (f) but different amplitudes (A1 and A2):
V1 / I1 = V2 / I2
where V1 and I1 are the voltage and current amplitudes, respectively, for the first sine wave, and V2 and I2 are the voltage and current amplitudes, respectively, for the second sine wave.
Since we have the same frequency for both waves, the frequency (f) cancels out in the equation, leaving us with:
V1 / I1 = V2 / I2
Now, if we rearrange the equation:
V1 / V2 = I1 / I2
This means that the ratio of the voltage amplitudes (V1 / V2) is equal to the ratio of the current amplitudes (I1 / I2). In other words, the ratio of the voltage amplitudes between two sine waves with equal frequencies but different amplitudes is the same as the ratio of their current amplitudes.
This relationship holds true for purely resistive circuits. If the circuit contains reactive elements like capacitors or inductors, the impedance will have a complex component, and the relationship between voltage and current amplitudes will be affected accordingly. In such cases, you'll need to consider the impedance and phase relationship between voltage and current.