Write the dimensions of the following in the terms of `"mass" , "time" , "length and charge"`
(a) Magnetic flux (b) Rigidity modulus.

Explanation

To find the dimensions of magnetic flux and rigidity modulus in terms of mass (M), time (T), length (L), and charge (Q), we will go through each term step by step.

### (a) Magnetic Flux

1. **Definition of Magnetic Flux**:
Magnetic flux (Φ) is defined as the product of the magnetic field (B) and the area (A) through which the field lines pass.
\[ \Phi = B \cdot A \]

2. **Dimensions of Area**:
The area (A) has dimensions of length squared.
\[ [A] = L^2 \]

3. **Magnetic Field Intensity**:
The magnetic field (B) can be expressed in terms of force (F), current (I), and length (L):
\[ B = \frac{F}{I \cdot L} \]

4. **Dimensions of Force**:
Force (F) has dimensions of mass times acceleration. Acceleration has dimensions of length per time squared.
\[ [F] = M \cdot \frac{L}{T^2} = M L T^{-2} \]

5. **Dimensions of Current**:
Current (I) is defined as charge per unit time.
\[ [I] = \frac{Q}{T} \]

6. **Substituting Dimensions into B**:
Now substituting the dimensions of force and current into the expression for B:
\[ [B] = \frac{M L T^{-2}}{\frac{Q}{T} \cdot L} = \frac{M L T^{-2} \cdot T}{Q \cdot L} = \frac{M T^{-1}}{Q} \]

7. **Combining Dimensions for Magnetic Flux**:
Now substituting the dimensions of B and A back into the equation for magnetic flux:
\[ [\Phi] = [B] \cdot [A] = \left(\frac{M T^{-1}}{Q}\right) \cdot (L^2) = M L^2 T^{-1} Q^{-1} \]

### Final Dimensions for Magnetic Flux:
\[ [\Phi] = M L^2 T^{-1} Q^{-1} \]

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### (b) Rigidity Modulus

1. **Definition of Rigidity Modulus**:
Rigidity modulus (G) is defined as the ratio of shear stress to shear strain.
\[ G = \frac{\text{Stress}}{\text{Strain}} \]

2. **Dimensions of Stress**:
Stress is defined as force per unit area.
\[ \text{Stress} = \frac{F}{A} \]

3. **Substituting Dimensions for Stress**:
Using the dimensions of force and area:
\[ [\text{Stress}] = \frac{[F]}{[A]} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]

4. **Dimensions of Strain**:
Strain is a dimensionless quantity (change in length/original length).
\[ [\text{Strain}] = 1 \]

5. **Combining Dimensions for Rigidity Modulus**:
Now substituting the dimensions of stress and strain into the equation for rigidity modulus:
\[ [G] = \frac{[\text{Stress}]}{[\text{Strain}]} = \frac{M L^{-1} T^{-2}}{1} = M L^{-1} T^{-2} \]

### Final Dimensions for Rigidity Modulus:
\[ [G] = M L^{-1} T^{-2} \]

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### Summary of Results:
- **Magnetic Flux**: \( [\Phi] = M L^2 T^{-1} Q^{-1} \)
- **Rigidity Modulus**: \( [G] = M L^{-1} T^{-2} \)

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