Introduction
In frequency modulation, the frequency of the carrier wave is varied in proportion to the amplitude of the modulating signal. The degree of variation in the frequency of the carrier wave is called the frequency deviation, and the ratio of the frequency deviation to the frequency of the modulating signal is called the modulation index.

Given Data
Carrier frequency (fc) = 100MHz = 100×10^6 Hz
Modulating frequency (fm) = 10 KHz = 10×10^3 Hz
Maximum frequency deviation (Δf) = 50 KHz = 50×10^3 Hz

Calculation of Modulation Index
The modulation index (m) can be calculated using the formula:

m = Δf / fm

Substituting the given values, we get:

m = 50×10^3 / 10×10^3
m = 5

Therefore, the modulation index is 5.

Calculation of Carrier Swing
The carrier swing is the total range of frequency variation of the carrier wave due to modulation. It can be calculated using the formula:

Carrier swing = 2Δf

Substituting the given values, we get:

Carrier swing = 2×50×10^3
Carrier swing = 100 KHz

Therefore, the carrier swing is 100 KHz.

Explanation
Frequency modulation involves varying the frequency of the carrier wave according to the amplitude of the modulating signal. The modulation index determines the degree of variation in the frequency of the carrier wave. A higher modulation index results in a greater degree of frequency variation, while a lower modulation index results in a smaller degree of frequency variation.

The carrier swing is the total range of variation in the frequency of the carrier wave due to modulation. It is twice the frequency deviation, as the carrier wave can vary in frequency by the same amount above and below the carrier frequency.

In this case, a 100 MHz carrier wave is frequency modulated by a 10 KHz modulating signal with a maximum frequency deviation of 50 KHz. The modulation index is calculated to be 5, indicating a relatively high degree of frequency variation. The carrier swing is calculated to be 100 KHz, indicating that the frequency of the carrier wave can vary by as much as ±50 KHz above and below the carrier frequency.