Question:
If \(A\), \(B\) and \(C\) represent work done, distance and electric charge respectively, then the physical quantity having the dimensions of
\[
\frac{C^2}{AB}
\]
is
If \(A\), \(B\) and \(C\) represent work done, distance and electric charge respectively, then the physical quantity having the dimensions of
\[
\frac{C^2}{AB}
\]
is
Show Hint
Remember the standard dimensions:
\[
[\varepsilon_0]
=
M^{-1}L^{-3}T^{4}I^{2},
\qquad
[\mu_0]
=
MLT^{-2}I^{-2}.
\]
Questions on dimensions are often solved quickly by matching these standard forms.
Updated On: Jul 9, 2026
- Permittivity
- Permeability
- Electric potential
- Electric energy
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The Correct Option is A
Solution and Explanation
Concept:
Write the dimensions of the given quantities:
\[
[A]=[\text{Work}]
=ML^2T^{-2}
\]
\[
[B]=[\text{Distance}]
=L
\]
\[
[C]=[\text{Charge}]
=IT
\]
Step 1: Find the dimensions of \(\dfrac{C^2}{AB}\). \[ \left[\frac{C^2}{AB}\right] = \frac{(IT)^2} {(ML^2T^{-2})(L)}. \] \[ = \frac{I^2T^2} {ML^3T^{-2}}. \] \[ = M^{-1}L^{-3}T^{4}I^{2}. \]
Step 2: Compare with known dimensions. From Coulomb's law, \[ F=\frac{1}{4\pi\varepsilon_0} \frac{q_1q_2}{r^2}. \] Hence \[ [\varepsilon_0] = \frac{Q^2} {F\,r^2}. \] Substituting dimensions, \[ [\varepsilon_0] = \frac{(IT)^2} {(MLT^{-2})(L^2)}. \] \[ = M^{-1}L^{-3}T^{4}I^{2}. \] This matches the dimensions obtained above. \[ \therefore \frac{C^2}{AB} \text{ has the dimensions of permittivity.} \]
Step 3: Write the final answer. \[ \boxed{\text{Permittivity}} \]
Step 1: Find the dimensions of \(\dfrac{C^2}{AB}\). \[ \left[\frac{C^2}{AB}\right] = \frac{(IT)^2} {(ML^2T^{-2})(L)}. \] \[ = \frac{I^2T^2} {ML^3T^{-2}}. \] \[ = M^{-1}L^{-3}T^{4}I^{2}. \]
Step 2: Compare with known dimensions. From Coulomb's law, \[ F=\frac{1}{4\pi\varepsilon_0} \frac{q_1q_2}{r^2}. \] Hence \[ [\varepsilon_0] = \frac{Q^2} {F\,r^2}. \] Substituting dimensions, \[ [\varepsilon_0] = \frac{(IT)^2} {(MLT^{-2})(L^2)}. \] \[ = M^{-1}L^{-3}T^{4}I^{2}. \] This matches the dimensions obtained above. \[ \therefore \frac{C^2}{AB} \text{ has the dimensions of permittivity.} \]
Step 3: Write the final answer. \[ \boxed{\text{Permittivity}} \]
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