Using dimensional analysis show that the unit for the time constant is seconds
Q=C*(delta)V
(delta)V=I*R
Q=R*C*I
Q=R*C*(Q/t)
1=R*C*(1/t)
RC=(1/(1/t))
so, what are the underlying units for R and C?
To show that the unit for the time constant is seconds using dimensional analysis, we will start with the formula for the time constant (τ), which is commonly used in physics and engineering for systems with exponential decay or growth:
τ = R * C
Where:
τ is the time constant,
R is the resistance, and
C is the capacitance.
To determine the units for the time constant, we need to examine the units of resistance and capacitance individually.
The unit for resistance is expressed in ohms (Ω), and the unit for capacitance is expressed in farads (F).
Now, let's break down the multiplication of the units of resistance and capacitance:
Ω * F = (kg * m^2 * s^-3 * A^-2) * (s^4 * A^2 * kg^-1 * m^-2)
By simplifying the units, we get:
Ω * F = kg * m^2 * s^-3 * s^4 * A^2 * A^-2 * kg^-1 * m^-2
Rearranging the terms, we have:
Ω * F = kg * kg^-1 * m^2 * m^-2 * s^-3 * s^4 * A^2 * A^-2
Simplifying further, we get:
Ω * F = s
Therefore, we have shown that the units of resistance (Ω) multiplied by the units of capacitance (F) result in the unit of time (seconds). Hence, the unit for the time constant (τ) is indeed seconds.