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Animated Solution for Physics - Electromagnetic Waves: If the magnetic field of a plane electromagnetic wave is given by $B = 100 \times 1 0^{- 6} sin [2 π \times 2 \times 1 0^{15} (t - \frac{x}{c})]$ then the maximum electric field associated with it is (Take, the speed of light $= 3 \times 1 0^{8}$ m/s)

Q: Why do we only need the amplitude and not the whole sine term?

The question asks for the *maximum* electric field ($E_{\max}$). The maximum value of the sine function is 1, so the maximum field is simply the amplitude ($B_{\max}$ or $E_{\max}$).

Q: Does the phase of the wave affect the maximum amplitude?

No, the phase $(\omega t - kx)$ determines the instantaneous value of the field at a specific time and position, but the maximum possible value (amplitude) remains constant.

Q: What if the wave was traveling in a medium other than a vacuum?

If the wave was in a medium, the speed of light $c$ would be replaced by the phase velocity $v$ in that medium, so the relation would be $E_{\max} = v B_{\max}$.

Select Answer:

Visualized Solution

$Extracting B_{m a x}$

Given magnetic field equation:
$B = 100 \times 1 0^{- 6} sin [2 π \times 2 \times 1 0^{15} (t - \frac{x}{c})]$
Comparing with standard equation $B = B_{m a x} sin (ω t - k x)$ :
$B_{m a x} = 100 \times 1 0^{- 6} T$

$Relation between E_{m a x} and B_{m a x}$

In an electromagnetic wave, the amplitudes of electric and magnetic fields are related by the speed of light $c$ .
$E_{m a x} = B_{m a x} \times c$
Given $c = 3 \times 1 0^{8} m/s$

$Calculating E_{m a x}$

$E_{m a x} = (100 \times 1 0^{- 6}) \times (3 \times 1 0^{8})$
$E_{m a x} = 300 \times 1 0^{2}$
$E_{m a x} = 3 \times 1 0^{4} N/C$

$Conclusion$

The maximum electric field associated with the wave is $3 \times 1 0^{4} N/C$ .
Correct option is (c).

$The Way Forward$

What if the wave was traveling in a medium with refractive index $n$ ?
The speed of light would be $v = \frac{c}{n}$ .
The relation becomes $E_{m a x} = v B_{m a x}$ .

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The Sigma Insight: Characteristics of Electromagnetic Waves

Solution Diagram

The problem asks us to find the maximum electric field of a plane electromagnetic wave, given the equation for its magnetic field. This is a classic application of the fundamental properties of electromagnetic waves.

Analyzing the Setup

We are given the instantaneous magnetic field equation:

B = 100 \times 1 0^{- 6} sin [2 π \times 2 \times 1 0^{15} (t - \frac{x}{c})]

To extract useful information, we compare this with the standard wave equation:

B = B_{m a x} sin (ω t - k x)

By direct comparison, the amplitude of the magnetic field, which is the maximum magnetic field

B_{m a x}

, is simply the coefficient of the sine function.

B_{m a x} = 100 \times 1 0^{- 6} T

The Master Equation

In any electromagnetic wave traveling through a vacuum, the electric and magnetic fields are intimately connected. They oscillate in phase, and their magnitudes are proportional to each other at every instant. The ratio of the amplitude of the electric field to the amplitude of the magnetic field is exactly equal to the speed of light,

c

This gives us our master equation:

E_{m a x} = B_{m a x} \times c

This elegant relationship allows us to instantly find one field if we know the other.

Final Calculation

We are given the speed of light

c = 3 \times 1 0^{8} m/s

. Now, we just need to substitute our values into the master equation:

E_{m a x} = (100 \times 1 0^{- 6} T) \times (3 \times 1 0^{8} m/s)

Let's group the numbers and the powers of ten:

E_{m a x} = (100 \times 3) \times (1 0^{- 6} \times 1 0^{8})

E_{m a x} = 300 \times 1 0^{2}

To write this in standard scientific notation, we adjust the decimal:

E_{m a x} = 3 \times 1 0^{4} N/C

The maximum electric field associated with the wave is $3 \times 1 0^{4} N/C$ . This perfectly matches option (c).

Common Traps & FAQs

Why do we only need the amplitude and not the whole sine term?

The question asks for the *maximum* electric field (

E_{m a x}

). The maximum value of the sine function is 1, so the maximum field is simply the amplitude (

B_{m a x}

E_{m a x}

Does the phase of the wave affect the maximum amplitude?

No, the phase

(ω t - k x)

determines the instantaneous value of the field at a specific time and position, but the maximum possible value (amplitude) remains constant.

What if the wave was traveling in a medium other than a vacuum?

If the wave was in a medium, the speed of light

c

would be replaced by the phase velocity

v

in that medium, so the relation would be

E_{m a x} = v B_{m a x}

Are the electric and magnetic fields in phase?

Yes, in a plane electromagnetic wave in a vacuum, the electric and magnetic fields oscillate perfectly in phase with each other.

Why are the units of E N/C while B is in Tesla?

Electric field is defined as force per unit charge (N/C or V/m), while magnetic field is defined by the Lorentz force law. The ratio

E / B

has units of velocity (m/s).

Select Answer:

Visualized Solution

$Extracting B_{m a x}$

$Relation between E_{m a x} and B_{m a x}$

$Calculating E_{m a x}$

$Conclusion$

$The Way Forward$

The Sigma Insight: Characteristics of Electromagnetic Waves

Analyzing the Setup

The Master Equation

Final Calculation

Similar Questions

The magnetic field of a plane electromagnetic wave is $B = 3 \times 1 0^{- 8} sin [200 π (y + c t)] \hat{i}$ T, where $c = 3 \times 1 0^{8}$ ms $^{- 1}$ is the speed of light. The corresponding electric field is

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If the magnetic field in a plane electromagnetic wave is given by $B = 3 \times 1 0^{- 8} sin (1.6 \times 1 0^{3} x + 48 \times 1 0^{10} t) \hat{j} T$ , then what will be expression for electric field?

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Extracting Bmax​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Relation between Emax​ and Bmax​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating Emax​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Way Forward

The Sigma Insight: Characteristics of Electromagnetic Waves

Analyzing the Setup

The Master Equation

Final Calculation

Similar Questions

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the magnetic field in a plane electromagnetic wave is given by B=3×10−8sin(1.6×103x+48×1010t)j^​ T, then what will be expression for electric field?

$Extracting B_{m a x}$

$Relation between E_{m a x} and B_{m a x}$

$Calculating E_{m a x}$

$Conclusion$

$The Way Forward$

The magnetic field of a plane electromagnetic wave is $B = 3 \times 1 0^{- 8} sin [200 π (y + c t)] \hat{i}$ T, where $c = 3 \times 1 0^{8}$ ms $^{- 1}$ is the speed of light. The corresponding electric field is

The electric field of a plane electromagnetic wave is given by $E = E_{0} (\hat{x} + \hat{y}) sin (k z - ω t)$ Its magnetic field will be given by

The electric field of a plane electromagnetic wave propagating along the x-direction in vacuum is $E = E_{0} \hat{j} cos (ω t - k x)$ . The magnetic field $B$ , at the moment $t = 0$ is

The electric field of a plane electromagnetic wave is given by $E = E_{0} \hat{i} cos (k z) cos (ω t)$ The corresponding magnetic field $B$ is then given by

If the magnetic field in a plane electromagnetic wave is given by $B = 3 \times 1 0^{- 8} sin (1.6 \times 1 0^{3} x + 48 \times 1 0^{10} t) \hat{j} T$ , then what will be expression for electric field?