Physics Lesson 16.12.4 - What is the Dimension of τ?

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Welcome to our Physics lesson on What is the Dimension of τ?, this is the fourth lesson of our suite of physics lessons covering the topic of RL Circuits, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

What is the Dimension of τ?

In the previous lesson, we gave the definition of the constant τ as the ratio of inductance L of the coil and resistance R of the resistor, i.e.

τ = L/R

Let's express the units of inductance and resistance in terms of the fundamental SI units and see what it gives. Thus, since

L = μ₀ ∙ N² ∙ A/I

we have for the unit of inductance L:

[Unit of L] = [N/A² ] ∙ [m² ]/[m]

Giving that

1N = 1 kg ∙ m/s²

we obtain

[Unit of L] = [kg ∙ m/s²/A² ] ∙ [m² ]/[m]
= [kg ∙ m²/A² ∙ s²]

As for the unit of resistance, we have from the Ohm's Law

[Unit of resistance] = [Unit of emf/Unit of current]
= [V/A]

By definition, emf is the work done to move the charges throughout the circuit. Thus,

1 [V] = 1[J/C]

Hence, since 1 C = 1 A ∙ s, we obtain

[Unit of resistance] = [J/C/A]
= [J/A ∙ C]
= [N ∙ m/A ∙ A ∙ s]
= [kg ∙ m/s² ∙ m/A² ∙ s]
= [kg ∙ m²/A² ∙ s³]

Therefore, we obtain for the unit of τ:

[Unit of τ] = [Unit of L/Unit of R]
= [kg ∙ m²/A² ∙ s²/kg ∙ m²/A² ∙ s³]
= [s]

Therefore, the unit of inductive time constant τ is the same as the unit of time, second [s] as expected (giving that it is a time constant). This is obvious, since the value of exponential part of the equation

i(t) = ε/R ∙ (1 - e^{- R ∙ t/L} )

must be dimensionless, in order to subtract it from 1.

You have reached the end of Physics lesson 16.12.4 What is the Dimension of τ?. There are 5 lessons in this physics tutorial covering RL Circuits, you can access all the lessons from this tutorial below.

More RL Circuits Lessons and Learning Resources

Magnetism Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
16.12	RL Circuits
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
16.12.1	What Are RL Circuits? An Overview
16.12.2	A Short Recap on RC Circuits as a Useful Step to Understand the RL Circuits
16.12.3	RL Circuits
16.12.4	What is the Dimension of τ?
16.12.5	What happens to the Current when the Source is removed from the Circuit

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