Question

Match column - I with column - II $$ \begin{array}{|l|l|} \hline \multicolumn{1}{|c|} {\text { Column - I }} & \multicolumn{1}{|c|} {\text { Column - II }} \\ \hline \text { (1) capacitance } & \text { (a) } \mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-3} \mathrm{~A}^{-1} \\ \hline \text { (2) Electricfield } & \text { (b) } \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1} \\ \hline \text { (3) planck's constant } & \text { (c) } \mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{4} \mathrm{~A}^{2} \\ \hline \text { (4) Angular momentum } & \text { (d) } \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1} \\ \hline \end{array} $$ (a) \(a, c, b, d\) (b) \(c, a, d, b\) (c) \(c, a, b, d\) (d) \(a, b, d, c\)

Short answer

The short answer is: (b) \(c, a, d, b\)

Step-by-step solution

Recall dimensional formulas for given quantities

We need to recall the dimensional formulas for each quantity: (1) Capacitance (C): \(\mathrm{[C] = \displaystyle\frac{Q}{V}}\) (2) Electric field (E): \(\mathrm{[E] = \displaystyle\frac{F}{q}}\) (3) Planck's constant (h): \(\mathrm{[h] = E \cdot T}\) (4) Angular momentum (L): \(\mathrm{[L] = I \cdot \omega}\)

Calculate dimensional formula for each quantity

First, we need to find the dimension of each quantity in terms of mass (M), length (L), time (T), and current (A) using the above formulas. (1) Capacitance (C): Using the formula, \(\displaystyle\frac{Q}{V} \rightarrow \frac{\mathrm{[ML^2T^{-3}A^{-1}]}{\mathrm{[LT^{-1}A^{-1}]}}\), we get \(\mathrm{[C] = M^{-1}L^{-2}T^{4}A^{2}}\) (2) Electric field (E): Using the formula, \(\displaystyle\frac{F}{q} \rightarrow \frac{\mathrm{[MLT^{-2}]}{\mathrm{[IT]}}\), we get \(\mathrm{[E] = M^{1}L^{1}T^{-3}A^{-1}}\) (3) Planck's constant (h): Using the formula, \(ET \rightarrow \mathrm{[ML^2T^{-2}]}\cdot\mathrm{[T]}\), we get \(\mathrm{[h] = M^{1}L^{2}T^{-1}}\) (4) Angular momentum (L): Using the formula, \(I\omega \rightarrow \mathrm{[ML^2]}\cdot\mathrm{[T^{-1}]}\), we get \(\mathrm{[L] = M^{1}L^{2}T^{-1}}\)

Match dimensional formulas with column II

Now, we need to match the dimension of each quantity `(1, 2, 3, 4)` with its respective dimensional formula `(a, b, c, d)` as follows: (1) Capacitance (C): \(\mathrm{[C] = M^{-1}L^{-2}T^{4}A^{2}}\) (c) (2) Electric field (E): \(\mathrm{[E] = M^{1}L^{1}T^{-3}A^{-1}}\) (a) (3) Planck's constant (h): \(\mathrm{[h] = M^{1}L^{2}T^{-1}}\) (d) (4) Angular momentum (L): \(\mathrm{[L] = M^{1}L^{2}T^{-1}}\) (b) So the correct matching is: (1) Capacitance → (c), (2) Electric field → (a), (3) Planck's constant → (d), and (4) Angular momentum → (b). Therefore, the correct answer is the option (b) \(c, a, d, b\).

Key concepts

Capacitance

Capacitance is a measure of a system's ability to store charge per unit of voltage. It is a fundamental concept in electromagnetism, and it helps to describe how much electric charge a capacitor can hold. Capacitors are electronic components that store electrical energy temporarily in an electric field.
When we look at the dimensional formula for capacitance, represented as \(\frac{Q}{V}\), where \(Q\) is charge and \(V\) is voltage, we understand how different physical quantities relate to each other. This expression can be expanded to dimensional formula units as \[M^{-1}L^{-2}T^{4}A^{2}\]. This notation breaks down as follows:

• \(M^{-1}\): Inverse of mass, indicating no dependence on mass in the physical sense.
• \(L^{-2}\): Inverse square relation with length, reflecting how capacitance also involves geometric arrangement.
• \(T^{4}\): In a power relation with time, influencing the time-dependent nature of charging and discharging.
• \(A^{2}\): Relates to the flow of electric current squared, representing the current's effect on electric fields.

Understanding these dimensions is crucial for circuits where the design and functionality hinge on correctly calculating and predicting capacitance.

Electric Field

The electric field is a vector field that surrounds charged particles and exerts force on other charged particles within the field. Its strength and direction can be visualized by the force a positive test charge would experience when placed in the field.
The dimensional formula for the electric field (\[\text{E}\]) appears as \[= \frac{F}{q}\], where \(F\) is force and \(q\) is electric charge. This expands to a dimensional formula of \[M^{1}L^{1}T^{-3}A^{-1}\]:

• \(M^{1}\): Indicates the dependency on mass, as force involves mass according to Newton's second law of motion.
• \(L^{1}\): Linear relation with length, as the force can vary directly with distance in an electric field.
• \(T^{-3}\): Exponential dependency on time, reflecting the inverse relation with time due to acceleration and velocity changes.
• \(A^{-1}\): Inverse dependency on electric current, associating with the current's influence in generating electric fields.

By understanding these dependencies, you can better predict the behavior of electric fields in various configurations such as around conductors or within capacitors.

Planck's Constant

Planck's constant is a fundamental constant in physics that characterizes the quantization of energy levels. It plays a crucial role in the field of quantum mechanics, describing the sizes of discrete energy packets (quanta) emitted or absorbed by atoms and molecules.
Expressed in terms of its dimensional formula as \[E \cdot T\] or energy multiplied by time, we arrive at the formulation \[M^{1}L^{2}T^{-1}\]:

• \(M^{1}\): Indicates mass dependency, due to its relation to energy (as in mass-energy equivalence).
• \(L^{2}\): Area relation with length, usually involving spatial dimensions in quantum mechanical frameworks.
• \(T^{-1}\): Inverse relation with time, as changes in time affect energy processes and their rates in quantum mechanics.

Planck’s constant essentially symbolizes the granularity in the universe at quantum scales, encapsulating the scale at which quantum effects become significant.

Angular Momentum

Angular momentum is a measure of the quantity of rotation of an object and is a key concept in both classical and quantum mechanics. It relates to how objects move when they spin or orbit in a system.
The dimensional formula of angular momentum, represented via \[I \cdot \omega\] (inertia times angular velocity), is given as \[M^{1}L^{2}T^{-1}\]:

• \(M^{1}\): Signifies dependency on mass, as inertia and rotational motion directly involve the mass of an object.
• \(L^{2}\): Square dependency on length, just like linear momentum but extended to rotational motion, revealing how radius or distance from the pivot influences angular momentum.
• \(T^{-1}\): Inverse proportionality with time, which is crucial for understanding how quickly rotations occur.

Angular momentum helps us analyze rotational dynamics, ensuring conservation laws apply not only in linear but also in circular and spinning systems.