Electric field in a plane electromagnetic wave is given by E = 50 sin(500x - 10 x 10^10 t) V/m. The

Q: Why do we use $v = \frac{\omega}{k}$ for the wave speed?

The phase of the wave is $(kx - \omega t)$. For a point of constant phase (like a crest), $kx - \omega t = \text{constant}$. Differentiating with respect to time gives $k \frac{dx}{dt} - \omega = 0$, which means the wave speed $v = \frac{dx}{dt} = \frac{\omega}{k}$.

Q: What does the coefficient $50$ represent in the equation?

The coefficient $50$ is the amplitude of the electric field, denoted by $E_0$. It represents the maximum value of the electric field in the wave, which is $50 \text{ V/m}$.

Q: Why is the speed of the wave less than $c$?

The wave is traveling in a medium, not in a vacuum. Electromagnetic waves interact with the atoms of the medium, which effectively slows down their propagation speed. The speed in a medium is always $v = \frac{c}{\mu}$, where $\mu$ is the refractive index.

Q: Does the negative sign in $(500x - 10 \times 10^{10} t)$ matter?

The negative sign indicates that the wave is propagating in the positive x-direction. If it were a positive sign, the wave would be traveling in the negative x-direction. However, it doesn't affect the magnitude of the wave speed.

Q: Can we find the magnetic field from this equation?

Yes! The amplitude of the magnetic field $B_0$ can be found using the relation $B_0 = \frac{E_0}{v}$. The magnetic field will oscillate in a plane perpendicular to both the electric field and the direction of propagation.

Question

Electric field in a plane electromagnetic wave is given by $E = 50 \sin(500x - 10 	imes 10^{10} t)$ V/m. The velocity of electromagnetic wave in this medium is
(Given, $c = $ speed of light in vacuum)

Options:

(A) $\frac{3}{2} c$

(B) $c$

(C) $\frac{2}{3} c$

(D) $\frac{c}{2}$

Sigma Percentile · Accepted Answer

The problem of finding the speed of an electromagnetic wave from its electric field equation is a classic and elegant exercise in wave mechanics. It tests our ability to read and interpret the mathematical language of waves. Let's dive into the thought process behind this solution.

### Decoding the Electromagnetic Wave

Imagine an electromagnetic wave traveling through a medium. The electric field of this wave is not static; it oscillates in space and time. The problem provides us with the exact mathematical description of this oscillation: 
$$E = 50 \sin(500x - 10 	imes 10^{10} t)$$

This equation might look intimidating at first glance, but it is actually packed with information. To unlock its secrets, we need to compare it with the standard equation of a traveling wave. The general form for a plane wave propagating in the positive x-direction is:
$$E = E_0 \sin(kx - \omega t)$$

By placing our given equation right next to the standard one, we can perform a direct term-by-term comparison. This is like matching a key to a lock.

### The Master Equation

From the comparison, we can immediately extract two critical parameters. First, the coefficient of $x$ is the wave number, denoted by $k$. So, we have:
$$k = 500 	ext{ rad/m}$$

Second, the coefficient of $t$ is the angular frequency, denoted by $\omega$. Thus, we find:
$$\omega = 10 	imes 10^{10} 	ext{ rad/s}$$

These two parameters, $k$ and $\omega$, are the DNA of the wave. They dictate how the wave behaves in space and time. But how do they relate to the speed of the wave?

### Calculating the Wave Speed

The speed $v$ of any traveling wave is fundamentally defined as the ratio of its angular frequency to its wave number. This is a powerful relation that you should always keep in your physics toolkit:
$$v = \frac{\omega}{k}$$

Now, it's just a matter of substituting the values we extracted. Let's plug them in:
$$v = \frac{10 	imes 10^{10}}{500}$$

To simplify this, we can write the numerator as $10^{11}$. Dividing $10^{11}$ by $500$ gives us:
$$v = 2 	imes 10^8 	ext{ m/s}$$

This is the actual speed of the electromagnetic wave as it travels through this specific medium. Notice that it is less than the speed of light in a vacuum, which makes perfect physical sense!

### The Final Connection

We have the speed $v$, but the options are given in terms of $c$, the speed of light in a vacuum. We know that $c$ is a universal constant:
$$c = 3 	imes 10^8 	ext{ m/s}$$

To find the relationship between $v$ and $c$, we simply take their ratio:
$$\frac{v}{c} = \frac{2 	imes 10^8}{3 	imes 10^8}$$

The $10^8$ terms cancel out beautifully, leaving us with a clean fraction:
$$\frac{v}{c} = \frac{2}{3}$$

Rearranging this gives us our final answer:
$$v = \frac{2}{3} c$$

This perfectly matches option (c). As a bonus thought, if you were asked for the refractive index $\mu$ of this medium, you could easily find it using $\mu = \frac{c}{v}$, which would be $\frac{3}{2}$ or $1.5$. Always look for these deeper connections in physics problems!

Animated Solution for Physics - Electromagnetic Waves: Electric field in a plane electromagnetic wave is given by $E = 50 sin (500 x - 10 \times 1 0^{10} t)$ V/m. The velocity of electromagnetic wave in this medium is (Given, $c =$ speed of light in vacuum)

Select Answer:

Visualized Solution

$E$ -field Equation

Standard Wave Equation

Extracting $k$ and $ω$

Wave Speed Formula

Substituting Values

Calculating Speed

Relating to $c$

Final Answer

The Way Forward

The Sigma Insight: Characteristics of Electromagnetic Waves

Decoding the Electromagnetic Wave

The Master Equation

Calculating the Wave Speed

The Final Connection

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