Question

The magnetic flux $\phi$ (in Wb) linked with a coil is related to time t (in s) as
$\phi = 5 At^2 + Bt - 2C$
The SI units of A and B are respectively

• A. Wbs$^2$, Wbs
• B. Wb s$^{-1}$, Wb
• C. Wbs$^{-2}$, Wbs$^{-1}$
• D. Wbs$^{-1}$, Wbs$^{-2}$

Hint:

Whenever an unknown physics coefficient is multiplied by time $t^n$ to match a specific unit $Y$, the unit of that coefficient is instantly and always $Y \times \text{s}^{-n}$.

Answer 1

The Correct Option is C

Concept:
• The rigorous Principle of Homogeneity of Dimensions constitutes a cornerstone of physics.

• It fundamentally states that in any physically meaningful and correct mathematical equation, every single term being added, subtracted, or equated must possess the exact same dimensional units.

• You absolutely cannot logically add apples to oranges; you cannot add volts to meters, nor can you subtract seconds from kilograms.

Step 1: Analyze the given mathematical equation
The provided time-dependent equation modeling the magnetic flux is:
\[ \phi = 5 At^2 + Bt - 2C \]
According to the problem statement, the unit of the resulting magnetic flux $\phi$ on the left side is exclusively Webers (Wb).
The unit of the variable time $t$ heavily featured on the right side is exclusively seconds (s).

Step 2: Apply the Principle of Homogeneity to the first term
Because the final outcome $\phi$ is measured strictly in Webers, the entirety of the first mathematical term, $5 At^2$, must also universally resolve to Webers.
The pure number 5 is completely dimensionless and irrelevant here.
Therefore, the dimensional unit of $(A \times t^2)$ must absolutely equal Wb.
\[ \text{Unit of } A \times (\text{s})^2 = \text{Wb} \]
Algebraically isolating the unit for the unknown coefficient A yields:
\[ \text{Unit of } A = \frac{\text{Wb}}{\text{s}^2} = \text{Wb s}^{-2} \]

Step 3: Apply the Principle of Homogeneity to the second term
Following the exact same strict logical reasoning, the entirety of the second mathematical term, $Bt$, must also definitively resolve to Webers.
Therefore, the dimensional unit of $(B \times t)$ must absolutely equal Wb.
\[ \text{Unit of } B \times (\text{s}) = \text{Wb} \]
Algebraically isolating the unit for the unknown coefficient B yields:
\[ \text{Unit of } B = \frac{\text{Wb}}{\text{s}} = \text{Wb s}^{-1} \]

Step 4: Conclusion
The rigorously derived SI units for the mathematical coefficients A and B are decisively Wb s$^{-2}$ and Wb s$^{-1}$ respectively. This perfectly and unquestionably matches option (C).