Show that the magnetic field due to a circular arc is given what
Answers
Answer:
Answer
the magnitude of the magnetic field in case of the circular arc is:
B= 4πRμ 0 i[θ]
where μ
0
permeability of medium in a vacuum, "r" is the radius of a circular arc, "i" is the current carrying by circular are and θ is the angle subtending at the centre.
So, magnetic field,
B=
4πR
μ
0
I
3
π
=
12R
μ
0
I
Circular wire is considered to be composed of small linear current elements. We determine magnetic field due to each of the linear current elements applying Biot-Savart law. Finally, we determine net magnetic field using superposition principle (i.e. by determining vector sum of magnetic fields due to all current elements).
In general, the bending of current carrying wire in circular shape has the effect of strengthening or localizing magnetic field in narrower region about the axis.
Direction of magnetic field (Right hand thumb rule)
Let us consider two diametrically opposite small current elements on the circular wire. The magnetic field lines are compressed inside the circle as it accommodates all the circular closed lines drawn outside. This compression of magnetic field lines is maximum at the center. In the figure here, we consider the circular coil in horizontal plane. The magnetic field lines being perpendicular to current elements are in the plane of drawing.
Magnetic field due to current in circular wire
Magnetic field lines due to oppositely placed current elements
Such is the case with any other pair of current elements as well. This means that magnetic field line passing though axis is reinforced by all such diametrically opposite pairs of current element. The magnetic field due to current in circular wire, therefore, is nearly axial.
The observations as above are the basis of Right hand thumb rule for current in circular wire. If we orient right hand such that curl of fingers follows the direction of current in the circular wire, then extended thumb points in the direction of magnetic field at its center.
Right hand thumb rule
If we orient right hand such that curl of fingers follows the direction of current in the circular wire, then extended thumb points in the direction of magnetic field at its center.
Right hand thumb rules for straight wire and circular wire are opposite in the notations. The curl of hand represents magnetic field in the case of straight wire, whereas it represents current in the case of circular wire. Similarly, the extended thumb represents current in the case of straight wire, whereas it represents magnetic field in the case of circular wire.
There is yet another simple way to find the direction of axial magnetic field at the center. Just look at the circular loop facing it. If the current is clockwise, then magnetic field is away from you and if the current is anticlockwise, then magnetic field is towards you.
Current in circular wire and magnet
The directional attributes of the magnetic field due to current in circular wire have an important deduction. If the current in a circular loop is anticlockwise when we look from one end (face), then the same current is clockwise when we look from opposite end (face). What it means that if direction of magnetic field is towards you from one face, then the direction of magnetic field is away from you from the other end and vice versa.
Directions of current in circular wir
Magnetic field due to current in circular wire
The magnitude of magnetic field due to a current element according to Biot-Savart law is given by :
đB=μ04πIđlsinθr2
But, θ=90° and sin90°=1. Also, r = R = Radius of circular wire.
⇒đB=μ04πIđlR2
All parameters except "đl" in the right hand expression of the equation are constants and as such they can be taken out of the integral.
B=∫đB=μ0I4πR2∫đl
The integration of dl over the complete circle is equal to its perimeter i.e. 2πR.
⇒B=μ0I4πR2X2πR=μ0I2R
If the wire is a coil having N circular turns, then magnetic filed at the center of coil is reinforced N times :
B=μ0NI2R