Currents in dc transmission lines can be 100 A or higher....

Currents in dc transmission lines can be 100 A or higher. Some people are concerned that the electromagnetic fields from such lines near their homes could pose health dangers. For a line that has current $150 \mathrm{~A}$ and a height of $8.0 \mathrm{~m}$ above the ground, what magnetic field does the line produce at ground level? Express your answer in teslas and as a percentage of the earth's magnetic field, which is $0.50 \mathrm{G}$. Is this value cause for worry?

Key Concepts

Comparative Analysis with Earth’s Magnetic Field

Assessing the significance of calculated magnetic fields often involves comparing them to the Earth’s own magnetic field. This comparison provides context regarding the magnitude of the artificial fields, helping determine whether they are likely to be strong enough to pose any health risks.

Unit Conversion in Magnetic Field Measurements

In electromagnetism, magnetic field strengths are commonly expressed in teslas (T) in the SI system and gauss (G) in other contexts, with 1 G equaling 10?? T. Understanding this conversion is essential for comparing different magnetic field values, such as contrived fields with natural phenomena like the Earth’s magnetic field.

Magnetic Field of a Current-Carrying Conductor

This concept involves understanding that a current flowing through a conductor produces a magnetic field around it, and the field’s magnitude decreases as the distance from the conductor increases. The quantitative calculation is typically done using the formula B = ??I/(2?r), which shows the inverse relation with distance and directly relates to the current in the conductor.

Biot–Savart Law and Ampère’s Law

These fundamental laws in electromagnetism provide the basis for calculating magnetic fields generated by moving electric charges. The Biot–Savart Law is used to compute the contribution of small segments of current to the overall magnetic field, while Ampère’s Law relates the total magnetic field circulation around a closed loop to the net current passing through it.

Transcript

00:01 Hi there.

00:03 So in the problem, it's given that the transmission lines can carry current of 100 ampers or more.

00:11 And the people who are leaving near the transmission lines are worried if the electromagnetic fields created because of the motion of current through this transmission line will have some health effects on them.

00:27 So let us explore.

00:28 Let us try to find the magnetic field.

00:31 Created because of this transmission line on the ground which is at a distance of 8 meters from the ground and let's see if it causes any danger the people living near the transmission lines so let's imagine that this is the ground level so this is your ground the g and let's say this is the transmission line which is at a distance of 8 meters above the ground.

01:07 It's given that some amount of current is flowing through this transmission line.

01:11 So the current that is flowing through the transmission line is 150 amps.

01:19 So the constant current of 150 amps is flowing through the transmission lines.

01:27 And so we are required to find the magnetic field on the ground because of the transmission line.

01:36 So i distance of 8 meters so this is a problem of a magnetic field near a long straight current carrying conductor and for a long straight current carrying conductor so if you have a long current carrying conductor if some current i is flowing through this conductor and if you want to find a magnetic field at this point which is at a distance of let's say r from this conductor then the magnetic field is given by equals new not which is the magnetic probability times the current divided by 2 pi and the distance at which you want to measure the magnetic field so this is the formula for the magnetic field because of a straight current carrying conductor at a distance are from the conductor so let's apply that formula here so we have our b equals mu not i divided by 2 pi r and uh so we know what is mu not it's the magnetic permeability of the free space and its value equals 4 pi times 10 rise to negative 7 times the current which is given to us 150 amps divided by again 2 pi times the distance at which we want to measure the magnetic field from the current carrying conductor and in our case the distance is 8 meters so i write 8 here so if you cancel this so you get 2 here so you have 300 divided by 8 times 10 rest 2 negative 7 and if you solve this answer comes out to be 37 .5 times 10 rest 2 negative 7 and the unit is tesla so this is the amount of magnetic field generated by so just remember it's negative 7 so this is the amount of magnetic field generated by the transmission line which is carrying 150 ampires of current at a ground at a distance of 8 meters from the line.

04:35 So 37 .5 times 10 raise 2 negative 7 tesla.

04:42 So it's given that the earth's magnetic field is 0 .5 gauss g.

04:58 And we know that the relation between the gauss and tesla is, so 1 gauss equals 10 raise to negative 4 tesla.

05:15 So if i convert the earth's magnetic field in the units of tesla, so i'm going to have 0 .5 times 10 raise 2 negative 4 tesla.

05:33 So this is my earth's magnetic field in terms of tesla...

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