Important Derivations for Class 12 Physics Chapter 7 Alternating Current

Important Derivations for Class 12 Physics Chapter 7 Alternating Current

Derivations Related to A.C. Applied Across an Inductor

Derivation 1:

(1) An ac voltage, V = V₀ sin wt, is applied across a pure inductor L. Obtain an expression for the current I in the circuit and hence obtain the
(i) inductive reactance of the circuit, and
(ii) the ‘phase’, of the current flowing, with respect to the applied voltage.

Solution:

the instantaneous ac potential difference across the ends of an inductor of inductance. $$ V=V_0 \sin \omega t $$ If I is the instantaneous current through L at instant t, $$ V=L \frac{d I}{d t} \text { or } V_0 \sin \omega t=L \frac{d I}{d t} $$ or $$d I=\frac{V_0}{L} \sin \omega t d t$$ Integrating both sides, $$ I=\frac{V_0}{L} \int_0^t \sin \omega t d t=\frac{V_0}{L}\left[\frac{-\cos \omega t}{\omega}\right]_0^t $$ or $$I=\frac{-V_0}{\omega L} \cos \omega t$ or $I=\frac{V_0}{\omega L} \sin \left(\omega t-\frac{\pi}{2}\right)$$ $$ I=I_0 \sin \left(\omega t-\frac{\pi}{2}\right) $$ where, $$I=\frac{V_0}{\omega L}$$ is the amplitude of the current.
(i) The quantity ωL in I = V0/ωL is analogous to the resistance and is called inductive reactance denoted by χ_L $$ X_L=\omega L=2 \pi \nu L $$ (ii) from above equations, it is clear that, in an ac circuit containing inductance, current lags voltage by π/2

Derivations Related to A.C. Applied Across a Capacitor

Derivation 2:

Show that the current leads the voltage in phase by π/2 in an ac circuit containing an ideal capacitor.

Solution:

Let us consider a capacitor C connected to an ac source as shown in the figure.

Let the ac voltage applied be V = V₀ sinwt

$$ \begin{aligned} & V=\frac{q}{C} \text { or } q=C V \\ & I=\frac{d q}{d t} \\ & I=\frac{d}{d t}\left(C V_0 \sin \omega t\right)=\omega C V_0 \cos \omega t=I_0 \cos \omega t \\ & I=I_0 \sin \left(\omega t+\frac{\pi}{2}\right) \end{aligned} $$ where, $$ I_0=\omega C V_0=\frac{V_0}{\frac{1}{\omega c}}=\text { current amplitude. } $$

Derivations Related to A.C. Applied Across Series LCR Circuit

Derivation 3:

An ac source of voltage V = V₀ sinwt is connected to a series combination of L, C and R. Use the phasor diagram to obtain expressions for impedance of the circuit and phase angle between voltage and current. Find the condition when current will be in phase with the voltage. What is the circuit in this condition called ?

Solution:

If I is the current in the circuit containing inductor of inductance L, capacitor of capacitance C and resistor of resistance R in series, then the voltage drop across the inductor is:
V_L = I × X_L
which leads current I by phase angle of π/2, and voltage drop across the capacitor is:

V_C = I × X_C

which lags behind current $I$ by phase angle of π/2, and voltage drop across the resistor is $$ V_R=I R $$ which in phase with current I. So, the net voltage E across the circuit is (using phasor diagram) $$ E=\sqrt{V_R^2+\left(V_L-V_C\right)^2} $$ or $$E=I \sqrt{R^2+\left(X_L-X_C\right)^2}$$ or, E=I Z where $$Z=\sqrt{R^2+\left(X_L-X_C\right)^2}$$ is known as impedance. Phase angle between voltage and current, is given by $$\tan \phi=\frac{V_L-V_C}{V_R}=\frac{X_L-X_C}{R}$$ A series LCR circuit has its natural angular frequency $$ \omega=\frac{1}{\sqrt{L C}} $$ and natural (resonating) frequency $$v=\frac{1}{2 \pi \sqrt{L C}}$$

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Why students face difficulty in physics derivations?

There can be several reasons why students face difficulty in physics derivations, including:

1. Lack of foundation: A lack of understanding of the underlying concepts and principles can make it difficult to follow the logical steps in a derivation.
2. Mathematical background: Physics often involves complex mathematical calculations, and students who struggle with math may find it challenging to perform the necessary calculations.
3. Limited practice: Regular practice is key to developing the skills required for physics derivations, and students who have limited practice opportunities may struggle.
4. Poor problem-solving skills: Physics problems often require creative problem-solving skills, and students who struggle with this aspect of physics may find derivations particularly challenging.
5. Limited exposure to different problems: Physics derivations can vary widely in their level of complexity, and students who have limited exposure to different types of problems may struggle when faced with a new challenge.
6. Confusion with notation: Physics often uses a specialized notation, and students who are unfamiliar with this notation may struggle to follow the steps in a derivation.

Remembering physics derivations can be a challenge, but here are some tips that may help:

1. Practice, practice, practice: Regularly practicing physics derivations helps to build muscle memory and improve recall.
2. Visualize the process: Try to visualize the steps involved in a derivation, as well as the physical meanings behind each equation.
3. Make connections: Try to relate each step of the derivation to a concept or formula you already know, which can help to reinforce your understanding.
4. Write it down: Writing out a derivation helps to solidify your understanding and makes it easier to remember.
5. Understand the physical meaning: Try to understand the physical meaning behind each equation, which can help you to remember the derivation in a broader context.
6. Teach someone else: Teaching someone else the derivation can be a great way to reinforce your understanding and remember it more easily.
7. Use mnemonics: Creating mnemonics or acronyms can be a helpful way to remember a series of steps in a derivation.

Remember, it takes time and consistent effort to remember physics derivations. Keep practicing and seeking help when needed, and you will likely see improvement over time.

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