A magnetic field can be produced by moving charges or electric currents(electric current is moving charged particles - electrons).
Biot-Savart law is the basic equation that gives us the magnetic field due to electric current in a conductor.
The magnetic field due to a current element dl at a point P with a distance r from dl is
Vector of dB = Vr(dB) = [1/4πε0c²]i[Vr(dl)×Vr(r)]/r³ ...(35.1)
where c = speed of light
i = current
Vr(dl) = length vector of the current element
Vr(r) = ector joining the current element to the point P where we are finding magnetic field
[1/ε0c²] is written as μ0 and is called the permeability of vacuum.
The value of μ0 is 4π*10-7
Vector of dB = Vr(dB) = [μ0i/4π][Vr(dl)×Vr(r)]/r³ ... (35.2)
The magnitude of the magnetic field is
dB = [μ0idl sinθ /4πr²]
Where θ is the angle between current element and the vector joining current element and the point P.
The direction of the field is perpendicular to the plane containing the current element and the point.
Magnetic field due to current in a straight wire at a point P with a distance d from it.
B = [μ0i/4πd][cosθ1 - cosθ2]
θ1 and θ2 are the values of θ corresponding to the lower end and the upper end respectively of the straigth wire.
See the Fig 35.3 to remember the method of measuring θ.
If point P is on a perpendicular bisector of the wire.
θ1 and θ2 are equal. If the length of the wire is α and distance of the point P is d from the wire
cosθ1 = α/SQRT(α²+ 4d²)
cosθ2 = - α/SQRT(α²+ 4d²)
θ1 is anlge withlower end of the wire.
θ2 is anle with the upper end of the wire.
B = [μ0iα]/[2πdSQRT(α²+ 4d²)]
If the straight wire is a very long one
θ1 = 0 and θ2 is equal to π. So
B = μ0i/2πd ... (35.6)
Magnetic field lines
Magnetic field lines are similar to electric field lines. A tangent to a magnetic field line gives the direction of the magnetic field existing at that point. For a long straight wire, the field lines are circles with their centres on the wire.
Force between parallel wires carryng current
If the two wires are treated as long straight wires carrying current i1 (W1) and i2 (W2)
Field on an element on the wire carrying current i2 is
B = μ0i1/2πd
Magnetic force on the element
dF = i2dlμ0i1/2πd
So the force per unit length of the wire
W2 due to the wire W1 is
dF/dl = i2μ0i1/2πd
= μ0i1i2/2πd ...(35.7)
Same amount of force is applied by W2 on unit length of W1.
If both the wires carry current in same direction they attract each other. If they carry current in opposite directions, they repel each other.
Definition of ampere
If two parallel, long wires, kept 1 m apart in vacuum, carry equal currents in the same directin and there is a force of attraction of 2*10-7 newton per metre of each wire, the current in each wire is said to be 1 ampere.
Field due to a circular current
Field at the centre
Radius of circular loop = a
current in the loop = i
B = μ0i/2a
Field at an axial point due to a circular conductor
B = μ0ia²/2(a²+d²)^(3/2)
a = radius of the circular conductor
d is the distance of the point from the centre of the circular conductor
If the field is far away from the centre
B = 2μ0πia²/4πd³
As πia² is magnetic dipole moment of circular conductor (μ)
B = 2μ0μ/4πd³ ... (35.9)
The circulation of Line integral of Vr(B).Vr(dl)of the resultant magnetic field along a closed, plane curve is equal to μ0 times the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant.
Line integral of Vr(B).Vr(dl) = μ0i ... (35.10)
You can derive the magnetic field at a point due to current in a long straight wire
using Ampere's law and verify that the formula is the same as the one derived by using Biot Savart Law.
Ampere's law can be derived from Biot-Savart law and Biot-Savart law from Ampere's law.
Ampere's law is useful under certain symmetrical conditions.
Finding magnetic field at a point due to a long, straight current using Ampere's law
A solenoid is an insulated wire wound closely in the form of a helix. The length of the solenoid is large compared to its radius of its loop.
The magnetic field inside a very tightly wound long solenoid is uniform everywhere and it zero outside it.
B = μ0ni ...(35.11)
n= number of turns per unit length
If a nonconducting ring is taken and a conducting wire is wound closely around it we get a toroid.
In a toroid magnetic field is
B = μ0Ni/2πr
Where N = number of turns in the toroid
r = is the distance of point P (where we have to find the magnetic field) from the centre of the toroid.