A radio tower is transmitting 30.0 kW of power equally in all directions. Assume that the radio

A radio tower is transmitting 30.0 kW of power equally in...

A radio tower is transmitting $30.0 \mathrm{~kW}$ of power equally in all directions. Assume that the radio waves that hit the Earth are reflected. a) What is the magnitude of the Poynting vector at a distance of $12.0 \mathrm{~km}$ from the tower? b) What is the root-mean-square value of the electric force on an electron at this location?

A radio tower is transmitting $30.0 \mathrm{~kW}$ of power equally in all directions. Assume that the radio waves that hit the Earth are reflected.
a) What is the magnitude of the Poynting vector at a distance of $12.0 \mathrm{~km}$ from the tower?
b) What is the root-mean-square value of the electric force on an electron at this location?

Key Concepts

Isotropic Radiation

Isotropic radiation refers to the emission of electromagnetic energy uniformly in all directions. In physics, sources that emit isotropically distribute energy equally over the surface of an expanding sphere, meaning that the energy per unit area decreases with the square of the distance from the source.

Inverse Square Law

The inverse square law is a fundamental principle stating that a physical quantity (such as intensity or force) is inversely proportional to the square of the distance from its source. When considering radiating energy from a point source, the energy spreads out over the surface area of a sphere, which increases as 4?r², leading to a decrease in intensity as distance increases.

Poynting Vector

The Poynting vector represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. It is a central concept in electromagnetism that quantifies how electromagnetic energy propagates through space, particularly relevant for calculating the intensity of waves at a given distance from the source.

Electric Field Amplitude and RMS Values

In electromagnetic waves, the electric field oscillates sinusoidally with time and space. The root-mean-square (rms) value of the electric field is used as a measure of its effective magnitude in relation to energy and forces, providing a convenient way to relate the field strength to observable phenomena such as the average power flow described by the Poynting vector.

Electric Force on a Charge

The electric force acting on a charged particle is determined by the product of the charge and the electric field it experiences. This relationship, derived from Coulomb's law, allows one to calculate the magnitude of force on an electron when the electric field is known, linking electromagnetic wave properties to the mechanical force on particles.

Transcript

00:01 Have an radio tower which is emitting power p equals to 30 .0 kilowatt equally in all directions.

00:07 So assuming that radio waves that hit the earth are reflected, so for the part a, we have to calculate the magnitude of the pointing vector as at a distance small r equals to 12 .0 kilometer.

00:20 Okay.

00:21 So the radiation emitted by the radio tower will be in the form of hemispair because it will reflect from the earth surface, so area of the hemispair, this will be equals to 2 pi r square.

00:34 Okay, so pointing vector as it can be written as power divided by area and it can be written as e squared by mu not multiplied by c.

00:44 Okay, so power can be calculated from here.

00:49 So pointing vector as can be calculated from here.

00:52 So as this will be equals to power p divided by area and this will be the area.

00:56 So, substituting 2 pi r square, so now we can substitute values.

01:00 So power p it is equals to 30 .0 multiplied by 10 to the power 3 watt divided by 2 pi multiplied by r which is 12 .0 multiplied by 10 to the power 3 meter whole square.

01:12 So from here after solving pointing vector s comes out to be 3 .32 multiplied by 10 to the power minus 5 watt per meter square.

01:21 So this is the answer for the part a.

01:23 Okay, now we can calculate the part b in which we have to calculate the root mean square value of the electric field.

01:30 So we have to calculate here erms.

01:33 So now we know that the power or pointing vector as this is equal to erms square divided by mu not multiplied by c...

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