Have you ever wondered how the electric field E and the magnetic field B are related? In electromagnetic theory, these two fields are two sides of the same coin. But finding their dimensional relationship might seem daunting if you try to write out the complex dimensional formulas for each of them from scratch.
The Lorentz Force Shortcut
Instead of memorizing the individual dimensions of E and B, we can use a powerful shortcut: The Lorentz Force.
Imagine a particle with charge q moving with a velocity v through a region of space that has both an electric field and a magnetic field. The particle will experience a force from both fields.
The electric force is given by:
Fe=qE
The magnetic force is given by:
Fm=qvB
Equating the Dimensions
By the
Principle of Dimensional Homogeneity, any two quantities that represent the same physical entity (in this case, force) must have the exact same dimensions. Therefore, the dimensions of the electric force must equal the dimensions of the magnetic force:
[Fe]=[Fm]
Substituting our formulas into this relation, we get:
[qE]=[qvB]
Notice how beautifully the charge
q appears on both sides! We can simply cancel it out, which leaves us with:
[E]=[v][B]
The Final Step
We know that velocity
v is simply distance divided by time. So, its dimensional formula is:
[v]=[L][T]−1
Substituting this back into our equation, we arrive at the final relationship:
[E]=[B][L][T]−1
And there you have it! The dimensions of the electric field are simply the dimensions of the magnetic field multiplied by the dimensions of velocity. This elegant result is a fundamental property of electromagnetic waves, where the ratio of the electric field to the magnetic field is exactly the speed of light, c!
