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Chapter 30: Problem 54

If at a certain instant, the magnetic induction of the electromagnetic wave in vacuum is \(6.7 \times 10^{-12} \mathrm{~T}\), then the magnitude of electric field intensity will be: (a) \(2 \times 10^{-3} \mathrm{~N} / \mathrm{C}\) (b) \(3 \times 10^{-3} \mathrm{~N} / \mathrm{C}\) (c) \(4 \times 10^{-3} \mathrm{~N} / \mathrm{C}\) (d) \(1 \times 10^{-3} \mathrm{~N} / \mathrm{C}\)

Short Answer

Expert verified
The magnitude of the electric field intensity is (a) 2 脳 10鈦宦� N/C.

Step by step solution

01

Identify the Relationship

The magnitude of the electric field intensity ( E ) and the magnetic induction ( B ) in a vacuum is related by the equation E = c imes B , where c is the speed of light in vacuum ( c = 3 imes 10^8 ext{ m/s} ). This relationship is derived from the properties of electromagnetic waves.
02

Substitute Known Values

Given that the magnetic induction ( B ) is 6.7 imes 10^{-12} ext{ T} and c = 3 imes 10^{8} ext{ m/s} , substitute these values into the equation E = c imes B to find the electric field intensity: E = 3 imes 10^{8} imes 6.7 imes 10^{-12} .
03

Calculate the Result

Multiply the numbers: 3 imes 10^{8} imes 6.7 imes 10^{-12} = 20.1 imes 10^{-4} = 2 imes 10^{-3} . Thus, the magnitude of the electric field intensity is 2 imes 10^{-3} ext{ N/C} .
04

Choose the Correct Option

Compare the calculated electric field intensity to the given options: (a) 2 imes 10^{-3} ext{ N/C} , (b) 3 imes 10^{-3} ext{ N/C} , (c) 4 imes 10^{-3} ext{ N/C} , (d) 1 imes 10^{-3} ext{ N/C} . The correct option is (a) 2 imes 10^{-3} ext{ N/C} .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Field Intensity
Electric field intensity is a measure of the force exerted by an electric field on a charged particle. It describes how strongly charged particles are pushed or pulled within that field. In mathematical terms, the electric field intensity \(E\) is expressed as the force \(F\) experienced per unit charge \(q\):\[ E = \frac{F}{q} \]
An important property of electromagnetic waves is that they propagate through a vacuum without the need for a medium. The electric field intensity of such a wave is directly proportional to the strength of the associated magnetic field, denoted as magnetic induction \(B\).
  • It's measured in newtons per coulomb (\(\mathrm{N/C}\)) or volts per meter (\(\mathrm{V/m}\)).
  • The electric field arises due to the change in the magnetic field over time, which is a fundamental characteristic of electromagnetic waves.
  • In the context of the exercise, the electric field intensity helps determine how an electromagnetic wave would interact with charged particles it encounters.
Magnetic Induction
Magnetic induction, often denoted as \(B\), is a measure of the magnetic field's ability to induce electric currents and forces in its surroundings. It represents the density of magnetic field lines, or magnetic flux, in a given area. The unit of magnetic induction is the tesla (\(\mathrm{T}\)).
In electromagnetic waves, magnetic induction works in concert with electric field intensity to propagate energy across space.
  • It reflects how much force is exerted by the magnetic aspect of an electromagnetic wave.
  • The relationship between electric field intensity and magnetic induction is critical, as demonstrated by the equation \(E = c \times B\), where \(c\) is the speed of light.
  • This relationship shows that the strength of the magnetic field directly influences the magnitude of the electric field within electromagnetic waves.
Speed of Light
The speed of light, represented by \(c\), is a fundamental constant of nature. Its value is approximately \(3 \times 10^8 \mathrm{~m/s}\) in a vacuum.
Light propagates as an electromagnetic wave, intertwining electric fields and magnetic fields that travel through space. The speed of light is central to the relationship between electric field intensity \(E\) and magnetic induction \(B\).
  • The equation \(E = c \times B\) highlights this relationship, showing how changes in the magnetic field relate to changes in the electric field as they travel through space.
  • The speed of light is crucial not only in physics but also in fields like telecommunications and astronomy, guiding our understanding of the universe.
  • In calculations involving electromagnetic waves, the speed of light serves as a bridge linking electricity and magnetism, consolidating them into a single framework.

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Most popular questions from this chapter

An output voltage of \(E=170 \sin 377 t\) is produced by an A.C. generator, where \(t\) is in sec, then the frequency of alternating voltage will be: (a) \(50 \mathrm{~Hz}\) (b) \(110 \mathrm{~Hz}\) (c) \(60 \mathrm{~Hz}\) (d) \(230 \mathrm{~Hz}\)

The maximum current in the circuit, if a capacitor of capacitance \(1 \mu \mathrm{F}\) is charged to a potential of \(2 \mathrm{~V}\) and is connected in parallel to an inductor of inductance \(10^{-3} \mathrm{H}\), is : (a) \(\sqrt{4000} \mathrm{~mA}\) (b) \(\sqrt{2000} \mathrm{~mA}\) (c) \(\sqrt{1000} \mathrm{~mA}\) (d) \(\sqrt{5000} \mathrm{~mA}\)

A TV tower has a height of \(100 \mathrm{~m}\). The area covered by the TV broadcast, if radius of the earth is \(6400 \mathrm{~km}\), will be : (a) \(380 \times 10^{7} \mathrm{~m}^{2}\) (b) \(402 \times 10^{7} \mathrm{~m}^{2}\) (c) \(595 \times 10^{7} \mathrm{~m}^{2}\) (d) \(440 \times 10^{7} \mathrm{~m}^{2}\)

An alternating voltage \(V=30 \sin 50 t+40 \cos 50 t\) is applied to a resistor of resistance \(10 \Omega\). The rms value of current through resistor is : (a) \(\frac{5}{\sqrt{2}} \mathrm{~A}\) (b) \(\frac{10}{\sqrt{2}} \mathrm{~A}\) (c) \(\frac{7}{\sqrt{2}} \mathrm{~A}\) (d) \(7 \mathrm{~A}\)

The peak and rms value of current in A.C. circuit. The current is represented by the equation \(i=5 \sin \left(300 t-\frac{\pi}{4}\right)\). where \(t\) is in seconds, and ' \(i^{\prime}\) in ampere : (a) \(5 \mathrm{~A}, 3.535 \mathrm{~A}\) (b) \(5 \mathrm{~A}, 5.53 \mathrm{~A}\) (c) \(3 \mathrm{~A}, 3.53 \mathrm{~A}\) (d) \(6.25 \mathrm{~A}, 5.33 \mathrm{~A}\)
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