Question

If \(B\) is magnetic induction, \(e\) is charge of electron, \(m\) is mass and \(c\) is speed of light in vacuum, then the physical quantity having dimensions of \[ \frac{4\pi mc}{Be} \] is

• A. Energy
• B. Electric potential
• C. Length
• D. Time

Hint:

Magnetic field dimensions are important: \[ [B]=MT^{-2}I^{-1} \] Use dimensional cancellation carefully.

Answer 1

The Correct Option is C

Step 1: Drop the pure number.
The factor $4\pi$ is dimensionless, so the dimensions of $\dfrac{4\pi mc}{Be}$ equal those of $\dfrac{mc}{Be}$.
Step 2: List dimensions of the symbols.
Mass $[m]=M$, speed $[c]=LT^{-1}$, charge $[e]=IT$, and magnetic field $[B]=MT^{-2}I^{-1}$ (from $F=BIL$).
Step 3: Build the numerator.
$[mc]=M\cdot LT^{-1}=MLT^{-1}$.
Step 4: Build the denominator.
$[Be]=(MT^{-2}I^{-1})(IT)=MT^{-1}$.
Step 5: Divide.
$\dfrac{[mc]}{[Be]}=\dfrac{MLT^{-1}}{MT^{-1}}=L$.
Step 6: Interpret.
A pure dimension of $L$ is a length, so among the options the quantity has the dimensions of length, option (3).
\[ \boxed{\text{Length}} \]