Question 15: Units , Dimensions and Error Analysis

Question

The dimensional formula of $\frac{1}{2} \epsilon_0 E^2$ ($\epsilon_0$ = permittivity of vacuum and $E$ = electric field) is $M^a L^b T^c$.
The value of $2a - b + c =$ ________.

Accepted Answer

For YoungÃ¢ÂÂs modulus of a wire,
$$ Y=\frac{FL}{A\,\Delta L} $$
For a circular wire, area
$$ A=\frac{\pi d^2}{4} $$
So,
$$ Y=\frac{4FL}{\pi d^2 \Delta L} $$
Since we have to find the error in $Y$, and error due to load is ignored, the fractional error is
$$ \frac{\Delta Y}{Y}=\frac{\Delta L}{L}+2\frac{\Delta d}{d}+\frac{\Delta (\Delta L)}{\Delta L} $$
Now substitute the given measurements.
Diameter:
$$ d=0.08\ 	ext{cm}, \qquad \Delta d=0.001\ 	ext{cm} $$
So,
$$ 2\frac{\Delta d}{d}=2	imes \frac{0.001}{0.08}=0.025 $$
Length:
$$ L=150\ 	ext{cm}, \qquad \Delta L=0.1\ 	ext{cm} $$
Hence,
$$ \frac{\Delta L}{L}=\frac{0.1}{150}=0.00067 $$
Extension:
$$ \Delta l=0.5\ 	ext{cm}, \qquad \Delta (\Delta l)=0.001\ 	ext{cm} $$
Thus,
$$ \frac{\Delta (\Delta l)}{\Delta l}=\frac{0.001}{0.5}=0.002 $$
Therefore total fractional error is
$$ \frac{\Delta Y}{Y}=0.00067+0.025+0.002=0.02767 $$
Now calculate $Y$.
Convert to SI units:
$$ L=150\ 	ext{cm}=1.5\ 	ext{m} $$
$$ d=0.08\ 	ext{cm}=8	imes 10^{-4}\ 	ext{m} $$
$$ \Delta l=0.5\ 	ext{cm}=5	imes 10^{-3}\ 	ext{m} $$
Given force,
$$ F=100\ 	ext{N} $$
Area:
$$ A=\frac{\pi d^2}{4} =\frac{\pi (8	imes 10^{-4})^2}{4} \approx 5.03	imes 10^{-7}\ 	ext{m}^2 $$
So,
$$ Y=\frac{FL}{A\Delta l} =\frac{100	imes 1.5}{(5.03	imes 10^{-7})(5	imes 10^{-3})} \approx 5.97	imes 10^{10}\ 	ext{N/m}^2 $$
Hence absolute error:
$$ \Delta Y=Y\left(\frac{\Delta Y}{Y}ight) $$
$$ \Delta Y=(5.97	imes 10^{10})(0.02767) \approx 1.65	imes 10^9\ 	ext{N/m}^2 $$
Therefore,
$$ \alpha=1.65 $$
So the correct answer is:
$$ \boxed{1.65} $$
Option B.

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Units , Dimensions and Error Analysis

The dimensional formula of \frac{1}{2} \epsilon_0 E^2 (\epsilon_0 = permittivity of vacuum and E = electric field) is M^a L^b T^c. The value of 2a - b + c = ________.

Choose the correct answer: