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R = L 2 π

The magnetic field due to current I in the circular wire is :

B C = μ 0 I 2 R = 2 π μ 0 I 2 L = π μ 0 I L = 3.14 μ 0 I L

In the case of straight wire, let us consider that wire is long enough for a point around middle of the wire. For comparison purpose, we assume that perpendicular linear distance used for calculating magnetic field due to current in straight wire is equal to the radius of circle. The magnetic field at a perpendicular distance “R” due to current in long straight wire is given as :

B L = μ 0 I 4 π R = μ 0 I X 2 π 4 π L = μ 0 I 2 L = 0.5 μ 0 I L

Clearly, the magnetic field due to current in circular wire is 6.28 times greater than that due to current in straight wire at comparable points of observations. Note that this is so even though we have given advantage to straight wire configuration by assuming it to be long wire. In a nutshell, a circular configuration tends to concentrate magnetic field along axial direction which is otherwise spread over the whole length of wire.

Problem : A current 10 A flowing through a straight wire is split at point A in two semicircular wires of radius 0.1 m. The resistances of upper and lower semicircular wires are 10 Ω and 20 Ω respectively. The currents rejoin to flow in the straight wire again as shown in the figure. Determine the magnetic field at the center “O”.

Magnetic field due to current in wire

Magnetic field due to current in wire

Solution : The straight wire sections on extension pass through the center. Hence, magnetic field due to straight wires is zero. Here, the incoming current at A is distributed in the inverse proportion of resistances. Let the subscripts “1” and “2’ denote upper and lower semicircular sections respectively. The two sections are equivalent to two resistances in parallel combination as shown in the figure. Here, potential difference between “A” and “B” is :

Currents in semicircular segments

Currents in semicircular segments

V A B = I X R 1 X R 2 R 1 + R 2 = I 1 R 1 = I 2 R 2 I 1 = I X R 2 R 1 + R 2 = 10 X 20 30 = 20 3 A I 2 = I X R 1 R 1 + R 2 = 10 X 10 30 = 10 3 A

We see that current in the upper section is twice that in the lower section i.e. I 1 = 2 I 2 . Also, the magnetic field is perpendicular to the plane of semicircular section (plane of drawing). The current in the upper semicircular wire is clockwise. Thus, the magnetic field due to upper section is into the plane of drawing. However, the current in the lower semicircular is anticlockwise. Thus, the magnetic field due to lower section is out of the plane of drawing. Putting θ = π for each semicircular section, the net magnetic field due to semicircular sections at “O” is:

B = μ 0 I 1 π 4 π R μ 0 I 2 π 4 π R B = μ 0 I 1 π 4 π R μ 0 I 1 π 8 π R = μ 0 I 1 π 8 π R B = 10 - 7 X 20 3 X 8 X 0.1 = 8.3 X 10 - 7 T

The net magnetic field is into the plane of drawing.

Exercises

An electron circles a single proton nucleus of radius 3.2 X 10 - 11 m with a frequency of 10 16 Hz. The charge on the electron is 1.6 X 10 - 19 Coulomb. What is the magnitude of magnetic field due to orbiting electron at the nucleus?

The equivalent current is given by :

I = q T = q ν

where ν and T are frequency and time period of revolutions respectively. The magnitude of magnetic field due to circular wire is given by :

B = μ 0 I 2 R

Substituting for I, we have :

B = μ 0 I 2 R = μ 0 q ν 2 R

Putting values,

B = 4 π 10 - 7 X 1.6 X 10 - 19 X 10 16 2 X 3.2 X 10 - 11 B = 31.4 T

Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
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Source:  OpenStax, Electricity and magnetism. OpenStax CNX. Oct 20, 2009 Download for free at http://cnx.org/content/col10909/1.13
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