Shortcut Methods and Tricks for Solving Numericals

Numerical-1: Potential Energy of Charges

Shortcut Method:

1. Remember the formula for potential energy ($PE = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r}$).
2. Substitute the values given in the question and calculate directly.

Explanation: The formula $PE = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r}$ takes care of all the constants and conversions, making the calculation straightforward. Simply plug in the values for the charges ($q_1 = +12 × 10^{−9}$ C, $q_2 = −6 × 10^{−9}$ C) and distance between them ($r = 60$ cm).

Tricks:

• For quick calculation, you can approximate the value of $\pi$ as 3.14 or 22/7.
• If the distance is given in centimeters, convert it to meters before plugging it into the formula.

Numerical-2: Electric Field at Corners of a Square

Shortcut Method:

1. Use the formula for electric field ($E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}$).
2. Calculate the distance from the point charge to each corner of the square using the Pythagorean theorem.
3. Substitute the values and calculate the electric field.

Explanation: The formula $E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}$ helps you determine the electric field due to a point charge. In this case, the key step is to find the distance from the charge to each corner. Since the square is symmetrical, all corners are at the same distance from the center. Applying the Pythagorean theorem, you can calculate this distance.

Tricks:

• If the side length of the square is given in centimeters, convert it to meters before calculating the distance.
• You can simplify the calculation by first finding the square of the distance and then taking the square root at the end.

Numerical-3: Capacitance and Charge of Parallel Plate Capacitor

Shortcut Method:

1. Remember the formula for capacitance ($C = \frac{\epsilon_0A}{d}$).
2. Substitute the given values and calculate the capacitance.
3. Use the formula for charge ($Q = CV$) to find the charge on the plates.

Explanation: For parallel plate capacitors, the formula $C = \frac{\epsilon_0A}{d}$ comes in handy. Simply substitute the values for area ($A = 0.1$ m²) and separation ($d = 1$ mm or 0.001 m). Then, use the formula $Q = CV$ to determine the charge on the plates, where $V$ is the given voltage (12 V).

Tricks:

• If the area is given in square centimeters (cm²), convert it to square meters (m²) by dividing by 10,000.
• If the separation is given in millimeters (mm), convert it to meters by dividing by 1,000.