Dimensions of `underset("electric flux")overset("magnetic flux") to` are

Options

A. `(L-T^(-1))`
B. `(TL^(-1))`
C. `(L^(3)r^(2)A^(-2))`
D. `(M^(0)L^(0)T^(0))`

Answer

To find the dimensions of the ratio of magnetic flux to electric flux, we will follow these steps:

### Step 1: Determine the dimension of magnetic flux
Magnetic flux (Φ_m) is given by the formula:
\[ Φ_m = B \cdot A \]
where \( B \) is the magnetic field and \( A \) is the area.

The dimension of area \( A \) is:
\[ [A] = L^2 \]

To find the dimension of the magnetic field \( B \), we can use the relationship between force, charge, and magnetic field:
\[ F = Q \cdot v \cdot B \]
Rearranging gives:
\[ B = \frac{F}{Q \cdot v} \]

The dimensions of force \( F \) are:
\[ [F] = M L T^{-2} \]

The dimension of charge \( Q \) is:
\[ [Q] = I \cdot T \]

The dimension of velocity \( v \) is:
\[ [v] = L T^{-1} \]

Substituting these into the equation for \( B \):
\[ [B] = \frac{M L T^{-2}}{(I \cdot T)(L T^{-1})} = \frac{M L T^{-2}}{I L T^{-1}} = \frac{M}{I T} \]

Now substituting back into the magnetic flux formula:
\[ [Φ_m] = [B] \cdot [A] = \left(\frac{M}{I T}\right) \cdot L^2 = \frac{M L^2}{I T} \]

### Step 2: Determine the dimension of electric flux
Electric flux (Φ_e) is given by:
\[ Φ_e = E \cdot A \]
where \( E \) is the electric field.

Using the relationship between force, charge, and electric field:
\[ F = Q \cdot E \]
Rearranging gives:
\[ E = \frac{F}{Q} \]

Substituting the dimensions:
\[ [E] = \frac{M L T^{-2}}{I T} = \frac{M L}{I T^2} \]

Now substituting back into the electric flux formula:
\[ [Φ_e] = [E] \cdot [A] = \left(\frac{M L}{I T^2}\right) \cdot L^2 = \frac{M L^3}{I T^2} \]

### Step 3: Calculate the ratio of magnetic flux to electric flux
Now we find the ratio:
\[ \frac{Φ_m}{Φ_e} = \frac{\frac{M L^2}{I T}}{\frac{M L^3}{I T^2}} \]

This simplifies to:
\[ \frac{Φ_m}{Φ_e} = \frac{M L^2}{I T} \cdot \frac{I T^2}{M L^3} = \frac{L^2 T^2}{L^3} = \frac{T^2}{L} = T^2 L^{-1} \]

### Final Answer
The dimensions of the ratio of magnetic flux to electric flux are:
\[ T^2 L^{-1} \]