Question 49: Units , Dimensions and Error Analysis

Question

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of $M^P L^Q T^R A^S$, where value of ' $Q$ ' and ' $R$ ' are

Accepted Answer

To solve this problem, we need to find the dimensions of the electric flux and the magnetic flux, and then take their ratio.
Determine the dimension of electric flux (ÃÂ¦Ã¢ÂÂ):
Electric flux is defined as
$$\Phi_E = \int_S \mathbf{E} \cdot d\mathbf{A}.$$
The electric field $\mathbf{E}$ has dimensions:
$$[E] = \frac{	ext{Force}}{	ext{Charge}}.$$
Since force has dimensions $$M L T^{-2},$$ and charge (in SI) has dimensions $$A\,T,$$ we get:
$$[E] = \frac{M L T^{-2}}{A\,T} = M L T^{-3} A^{-1}.$$
The area element $$dA$$ has dimensions $$L^2.$$
Thus, the electric flux has dimensions:
$$[\Phi_E] = [E][A] = M L T^{-3} A^{-1} 	imes L^2 = M L^3 T^{-3} A^{-1}.$$
Determine the dimension of magnetic flux (ÃÂ¦_B):
Magnetic flux is defined as
$$\Phi_B = \int_S \mathbf{B} \cdot d\mathbf{A}.$$
The magnetic field $\mathbf{B}$ has dimensions determined via the Lorentz force law (using $$F = q\,v\,B$$), giving:
$$[B] = \frac{F}{q\,v} = \frac{M L T^{-2}}{(A\,T)(L T^{-1})} = M T^{-2} A^{-1}.$$
Again, the area element has dimensions $$L^2.$$
So the magnetic flux has dimensions:
$$[\Phi_B] = [B][A] = (M T^{-2} A^{-1})(L^2) = M L^2 T^{-2} A^{-1}.$$
Find the ratio of electric flux to magnetic flux:
Taking the ratio:
$$\frac{\Phi_E}{\Phi_B} = \frac{M L^3 T^{-3} A^{-1}}{M L^2 T^{-2} A^{-1}}.$$
Canceling like terms:
Mass $(M)$ cancels.
Current $(A^{-1})$ cancels.
For length, $$L^{3-2} = L^1.$$
For time, $$T^{-3 -(-2)} = T^{-1}.$$
Therefore, the ratio has dimensions:
$$L^{1}T^{-1}.$$
This means in the expression
$$M^P L^Q T^R A^S,$$
the exponents for length and time are $$Q = 1$$ and $$R = -1.$$
Looking at the options provided, this corresponds to Option D: $$(1,-1).$$
Thus, the answer is Option D.

Answer

$(3,-5)$

Answer

$(-2,1)$

Answer

$(-2,2)$

Answer

$(1,-1)$

Units , Dimensions and Error Analysis

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of M^P L^Q T^R A^S, where value of ' Q ' and ' R ' are

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