Question

Как изменится период свободных электромагнитных колебаний в контуре, если ёмкость конденсатора уменьшить в 2 раза, а индуктивность катушки увеличить в 8 раз?

Answer 1

Changes in the Period of Free Electromagnetic Oscillations in a Circuit

When the capacitance of a capacitor is reduced by a factor of 2 and the inductance of a coil is increased by a factor of 8, the period of free electromagnetic oscillations in the circuit will change. To understand how the period changes, let's consider the formula for the period of oscillation in an LC circuit:

T = 2π√(LC)

Where: - T is the period of oscillation - π is a mathematical constant approximately equal to 3.14159 - L is the inductance of the coil - C is the capacitance of the capacitor

To determine the effect of changing the capacitance and inductance on the period, we can substitute the new values into the formula and compare the results.

Effect of Reducing Capacitance by a Factor of 2

If the capacitance of the capacitor is reduced by a factor of 2, we can represent the new capacitance as C' = C/2. Substituting this value into the formula, we get:

T' = 2π√(L(C/2))

Simplifying the equation, we have:

T' = 2π√(LC)/√2

Therefore, reducing the capacitance by a factor of 2 will result in the period of oscillation being multiplied by √2.

Effect of Increasing Inductance by a Factor of 8

If the inductance of the coil is increased by a factor of 8, we can represent the new inductance as L' = 8L. Substituting this value into the formula, we get:

T' = 2π√((8L)C)

Simplifying the equation, we have:

T' = 2π√(8LC)

Since √(8LC) = 2√(LC), we can further simplify the equation to:

T' = 4π√(LC)

Therefore, increasing the inductance by a factor of 8 will result in the period of oscillation being multiplied by 4.

Overall Effect on the Period of Oscillation

To determine the overall effect on the period of oscillation, we need to consider the combined effect of both changes.

From the previous calculations, we know that reducing the capacitance by a factor of 2 multiplies the period by √2, and increasing the inductance by a factor of 8 multiplies the period by 4. Therefore, the overall effect can be calculated as:

T' = (4π√(LC)) * √2

Simplifying the equation, we have:

T' = 4π√(2LC)

Therefore, when the capacitance of the capacitor is reduced by a factor of 2 and the inductance of the coil is increased by a factor of 8, the period of free electromagnetic oscillations in the circuit will be multiplied by approximately 4π√2.

Please note that the above calculations are based on the assumption that other factors, such as resistance, remain constant in the circuit.