# 0.4 Magnetic field due to current in a circular wire  (Page 3/5)

 Page 3 / 5

Problem : Calculate magnetic field at the center O for the current flowing through wire segment as shown in the figure. Here, current through wire is 10 A and radius of the circular part is 0.1 m.

Solution : Magnetic field at O is contributed by long straight wire and circular wire. The direction of magnetic field at O due to straight part of the wire is into the plane of drawing as obtained by applying Right hand thumb rule for straight wire. The direction of current in the circular part is anticlockwise and hence magnetic field due to this part is out of the plane of drawing as obtained by applying Right hand thumb rule for circular wire.

The magnitude of magnetic field due to circular wire is :

${B}_{C}=\frac{{\mu }_{0}I}{2R}$

The magnitude of magnetic field due to straight wire is :

${B}_{S}=\frac{{\mu }_{0}I}{2\pi R}$

Hence, magnitude of magnetic field at O is algebraic sum of two magnetic fields (we consider outward direction as positive) :

$B={B}_{C}-{B}_{S}=\frac{{\mu }_{0}I}{2R}-\frac{{\mu }_{0}I}{2\pi R}$

Putting values :

$⇒B=\frac{4\pi {10}^{-7}X10}{2X0.1}-\frac{4\pi {10}^{-7}X10}{2\pi X0.1}$ $⇒B=62.9X{10}^{-6}-20X{10}^{-6}=42.9\phantom{\rule{1em}{0ex}}\mu T$

The net magnetic field is acting out of the plane of paper.

## Magnitude of magnetic field due to current in circular arc

The magnitude of magnetic field due to current in arc shaped wire can be obtained by integrating Biot-Savart expression in an appropriate range. Now, the integral set up for current in circular wire is :

$B=\int đB=\frac{{\mu }_{0}I}{4\pi {R}^{2}}\int đl$

Circular arc is generally referred in terms of the angle θ, it subtends at the center of the circle. From geometry, we know that :

$đl=Rđ\theta$ Substituting in the integral and taking the constant R out of the integral, we have :

$B=\frac{{\mu }_{0}I}{4\pi R}\int đ\theta$ $⇒B=\frac{{\mu }_{0}I\theta }{4\pi R}$

This is the expression for the magnitude of magnetic field due to current in an arc which subtends an angle θ at the center. Note that the expression is true for the circle for which θ = 2π and magnetic field is :

$⇒B=\frac{{\mu }_{0}IX2\pi }{4\pi R}=\frac{{\mu }_{0}I}{2R}$

Problem : Find the magnetic field at the corner O due to current in the wire as shown in the figure. Here, radius of curvature is 0.1 m for the quarter circle arc and current is 10 A.

Solution :

Here the straight line wire segment AB and CD when extended meet at O. As such, there is no magnetic field due to current in these segments. The magnetic field at O is, therefore, solely due to magnetic field due to quarter arc AC. The arc subtends an angle π/2 at its center i.e O. Now,

$B=\frac{{\mu }_{0}I\theta }{4\pi R}$

Putting values, we have :

$⇒B=\frac{{10}^{-7}X10X\pi }{0.1X2}=0.157X{10}^{-6}=0.157\phantom{\rule{1em}{0ex}}\mu T$

Since current in the arc is anticlockwise, magnetic field is perpendicular and out of the plane of drawing.

## Current in straight wire .vs. current in circular wire

A length of wire, say L, is given and it is asked to maximize magnetic field in a region due to a current I in the wire. Which configuration would we consider – a straight wire or a circular wire? Let us examine the magnetic fields produced by these two configurations.

If we bend the wire in the circle, then the radius of the circle is :

#### Questions & Answers

how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-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
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 A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Got questions? Join the online conversation and get instant answers!