Derive the mathematical expression for electric field intensity and solve problems related to parallel capacitors.

$$
E = \boxed{\frac{1}{4\pi\epsilon_0} \frac{q}{r^2}}
$$

b)
Given: $C_1 = 10 \mu\text{F}$, $C_2 = 20 \mu\text{F}$, $V = 90 \text{ V}$.

a) Calculate the equivalent capacitance of the combination.

b) Calculate the charge stored on each capacitor.

Use the formula $Q = CV$.
For $C_1$:

For $C_2$:

Q.15:

a)

• Ampere's Law: Ampere's Law states that the line integral of the magnetic field $\vec{B}$ around any closed loop is equal to $\mu_0$ times the total current $I_{enc}$ passing through the loop. $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$
• Magnetic Field Inside an Ideal Solenoid:

The magnetic field inside is uniform and parallel to the axis, and zero outside.

The line integral $\oint \vec{B} \cdot d\vec{l}$ simplifies to $BL$ because the field is uniform and parallel to AB, and zero or perpendicular along other segments. $\oint \vec{B} \cdot d\vec{l} = BL$

If there are $n$ turns per unit length, then in length $L$, there are $nL$ turns. Each turn carries current $I$. $I_{enc} = nLI$

$$
B = \boxed{\mu_0 n I}
$$

b)
The four factors that affect the magnitude of the force acting on a charged particle moving in a magnetic field are:

• The magnitude of the charge ($q$) of the particle.
• The magnitude of the velocity ($v$) of the particle.
• The magnitude of the magnetic field strength ($B$).
• The angle ($\theta$) between the velocity vector of the particle and the magnetic field vector.

Q.16:

a)

• Condition for Resonance: In an RLC series AC circuit, resonance occurs when the inductive reactance ($X_L$) is equal to the capacitive reactance ($X_C$). $$X_L = X_C$$
• Formula for Resonant Frequency:

$$
f_0 = \boxed{\frac{1}{2\pi\sqrt{LC}}}
$$

• Impedance at Resonance:

$$
Z = \sqrt{R^2 + 0^2}
$$
$$
Z = \boxed{R}
$$

b)
Given: $V_{rms} = 200 \text{ V}$, $I_{rms} = 0.5 \text{ A}$, power factor $\cos\phi = 0.6$.