The energy $U$ stored in an inductor carrying a current $I$ is related to the magnetic flux $\Phi_B$ by the expression $U = \frac{1}{2} \Phi_B I$. What is the dimensional formula of magnetic flux ($\Phi_B$)?
$ML^2T^{-2}A^{-1}$
Dimensional Formula Calculation for Magnetic Flux ($\Phi_B$)
This solution explains how to determine the dimensional formula for magnetic flux ($\Phi_B$) using the provided relationship between energy ($U$), current ($I$), and magnetic flux ($\Phi_B$). We are given the expression relating these quantities for an inductor:
$$U = \frac{1}{2} \Phi_B I$$
Our goal is to find the dimensions of magnetic flux, $[\Phi_B]$. To do this, we first need to know the dimensions of energy ($U$) and current ($I$).
Deriving Magnetic Flux Dimensions from the Energy Expression
We can rearrange the given formula to isolate the magnetic flux ($\Phi_B$):
$$ \Phi_B = \frac{2U}{I} $$
Now, we can find the dimensions of $\Phi_B$ by dividing the dimensions of energy ($U$) by the dimensions of current ($I$). The factor $\frac{1}{2}$ is a dimensionless constant and does not affect the dimensional formula.
The dimensional formula for magnetic flux, $[\Phi_B]$, can be expressed as:
$$ [\Phi_B] = \frac{[U]}{[I]} $$
Analysis of Energy and Current Dimensions
Let's determine the dimensions of the quantities involved:
- Energy ($U$): Energy is dimensionally equivalent to work. Work is defined as force multiplied by distance. The dimensions of force are mass ($M$) times acceleration ($LT^{-2}$), and the dimension of distance is length ($L$). Therefore, the dimensions of energy are: $$ [U] = [\text{Force}] \times [\text{Distance}] = (MLT^{-2}) \times (L) = ML^2T^{-2} $$
- Current ($I$): Electric current is considered a fundamental quantity in physics and its dimension is represented by $A$ (Ampere). $$ [I] = A $$
Calculating the Final Dimensional Formula
Substitute the dimensions of energy and current into the rearranged formula for magnetic flux:
$$ [\Phi_B] = \frac{ML^2T^{-2}}{A} $$
Expressing this with a negative exponent for current, we get:
$$ [\Phi_B] = ML^2T^{-2}A^{-1} $$
This result represents the dimensional formula for magnetic flux ($\Phi_B$). Comparing this with the given options, we find that it matches one of them.
Summary of Dimensions:
| Quantity | Symbol | Dimensional Formula |
|---|---|---|
| Energy | $U$ | $ML^2T^{-2}$ |
| Current | $I$ | $A$ |
| Magnetic Flux | $\Phi_B$ | $ML^2T^{-2}A^{-1}$ |
Therefore, the dimensional formula for magnetic flux is $ML^2T^{-2}A^{-1}$.
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