6. Magnetic Field Due To A Current By Sanjay Pandey

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6. Magnetic Field due to a Current

Biot-Savart law

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 𝑑𝑑𝑑𝑑 at a point P with a distance 𝑟𝑟 from 𝑑𝑑𝑑𝑑 is 1 𝑖𝑖(𝑑𝑑𝑙𝑙⃗×𝑟𝑟⃗) �⃗ = 𝑑𝑑𝐵𝐵 2 3 4𝜋𝜋𝜀𝜀 0 𝑐𝑐

∵

∴

𝑟𝑟

Where 𝑐𝑐 = speed of light 𝑖𝑖 = current 𝑑𝑑𝑙𝑙⃗ = length vector of the current element 𝑟𝑟⃗ = vector joining the current element to the point P where we are finding magnetic field 1 2 = is written as 𝜇𝜇0 and is called the permeability of vacuum. 𝜀𝜀 0 𝑐𝑐

The value of 𝜇𝜇0 is 4π×10−7 𝜇𝜇 0

�⃗ = 𝑑𝑑𝐵𝐵

4𝜋𝜋

𝑑𝑑𝑑𝑑 =

4π

×

𝑖𝑖𝑖𝑖 𝑙𝑙⃗×𝑟𝑟⃗ 𝑟𝑟 3

The magnitude of the magnetic field is 𝜇𝜇 0 𝑖𝑖𝑖𝑖𝑖𝑖 sin 𝜃𝜃 𝑟𝑟 2

where 𝜃𝜃 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. 2.

Magnetic field due to current in a straight wire at a point P with a distance d from it 𝐵𝐵 =

𝜇𝜇 0 𝑖𝑖

4𝜋𝜋𝜋𝜋

[cos 𝜃𝜃1 − cos 𝜃𝜃2 ]

𝜃𝜃1 and 𝜃𝜃2 are the values of 𝜃𝜃 corresponding to the lower end and the upper end respectively of the straigth wire.

If point P is on a perpendicular bisector of the wire, then θ1 and θ2 are equal. If the length of the wire is α and distance of the point P is d from the wire cos 𝜃𝜃1 = 𝛼𝛼/�(𝛼𝛼² + 4𝑑𝑑²)

cos 𝜃𝜃2 = − 𝛼𝛼/�(𝛼𝛼² + 4𝑑𝑑²)

𝜃𝜃1 is anlge withlower end of the wire. 𝜃𝜃2 is anle with the upper end of the wire.

2

𝐵𝐵 =

𝜇𝜇0 𝑖𝑖𝑖𝑖

2𝜋𝜋𝜋𝜋�(𝛼𝛼 2 + 4𝑑𝑑 2 )

If the straight wire is a very long one 𝜃𝜃₁ = 0 and θ₂ is equal to π. So

𝐵𝐵 = 𝜇𝜇₀𝑖𝑖/2𝜋𝜋𝜋𝜋

3.

Magnetic field lines

4.

Rule to Find Direction of Field Around a Wire

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Maxwell's Screw Rule

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 centers on the wire. The following two rules are used to predict the direction of the magnetic field around the wires in different situations. If a right-handed screw is turned so that, it moves forwards in the same direction as an electric current, its direction of rotation gives the direction of the magnetic field due to the current.

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The Right Hand Grip Rule

If a wire carrying a current is gripped with the right hand and with the thumb pointing along the wire in the direction of the current, then the fingers point in the direction of the magnetic field around the wire. 5.

Force between parallel wires carryng current

If the two wires 1 and 2 are treated as long straight wires carrying current i₁ and i₂ respectively, then

Field on an element on the wire carrying current i₂ is 𝐵𝐵 = 𝜇𝜇₀𝑖𝑖₁/2𝜋𝜋𝜋𝜋

Magnetic force on the element

𝑑𝑑𝑑𝑑 = 𝑖𝑖2 𝑑𝑑𝑑𝑑

𝜇𝜇 0 𝑖𝑖1 2𝜋𝜋𝜋𝜋

3 So the force per unit length of the wire 2 due to the wire 1 is 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

=

𝑖𝑖₂𝜇𝜇 ₀𝑖𝑖₁ 2𝜋𝜋𝜋𝜋

=

𝜇𝜇 ₀𝑖𝑖₁𝑖𝑖₂ 2𝜋𝜋𝜋𝜋

Same amount of force is applied by 2 on unit length of 1.

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. 6. •

Field due to a circular current Field at the centre Radius of circular loop = 𝑎𝑎

current in the loop = 𝑖𝑖 𝐵𝐵 =

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𝜇𝜇 ₀𝑖𝑖 2𝑎𝑎

Field at an axial point due to a circular conductor

𝐵𝐵 =

𝜇𝜇 ₀𝑖𝑖𝑎𝑎 2

3

2(𝑎𝑎 2 +𝑥𝑥 2 )2

where 𝑎𝑎 = radius of the circular conductor 𝑥𝑥 is the distance of the point from the centre of the circular conductor

If the field is far away from the centre i.e., 𝑥𝑥 >> 𝑎𝑎 then

𝐵𝐵 =

2𝜇𝜇 0 𝜋𝜋𝜋𝜋 𝑎𝑎 2 4𝜋𝜋𝑥𝑥 3

As 𝜋𝜋𝜋𝜋𝜋𝜋² is magnetic dipole moment of circular conductor (𝜇𝜇⃗ )

7.

Ampere's law

�⃗ = 2𝜇𝜇0 𝜇𝜇⃗ /4𝜋𝜋𝜋𝜋³ 𝐵𝐵

4 ���⃗ of the resultant magnetic field along a closed, plane curve is �⃗. 𝑑𝑑𝑑𝑑 The circulation of Line integral of 𝐵𝐵 equal to 𝜇𝜇0 times the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant. H

dl

�⃗. ���⃗ ∮ 𝐵𝐵 dl = μ0 𝑖𝑖

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. �⃗ satisfy the right-hand rule. The sign convention for the direction of C is taken so that I and 𝐵𝐵

The right hand rule: Curl your right-hand fingers around the closed path (Amperian loop), with them pointing in the direction of integration. A current passing through the loop in the general direction of your outstretched thumb is assigned a plus sign, and a current in the opposite direction is assigned a minus sign. 8. • • • • • 9. •

Guidelines to use Amperes circuital law If B is everywhere tangent to the integration path and has the same magnitude B at every point on the path, then its line integral is equal to B multiplied by the circumference of the path. If B is everywhere perpendicular to the path, for all or some portion of the path, that portion of the path makes no contribution to the line integral. ���⃗ 'B' is always the total magnetic field at each point on the path. In general, this �⃗. dl In the integral , ∮ 𝐵𝐵 field is caused partly by currents linked by the path and partly by the currents outside. Even when no current is linked by the path, the field at points on the path need not be zero. Two useful guiding principles to choose an integration path are that the point at which the field is to be determined must lie on the path and that the path must have enough symmetry so that the integral can be valuated easily. 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 Solenoid

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. 𝐵𝐵 = 𝜇𝜇0 𝑛𝑛𝑛𝑛

𝑛𝑛 = number of turns per unit length

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Toroid

If a non conducting ring is taken and a conducting wire is wound closely around it we get a toroid.

In a toroid magnetic field is Where

𝐵𝐵 =

𝜇𝜇 0 𝑁𝑁𝑁𝑁 2𝜋𝜋𝜋𝜋

𝑁𝑁 = number of turns in the toroid

𝑟𝑟 = is the distance of point P (where we have to find the magnetic field) from the centre of the toroid.

10. Dipole Moment of Current Loop Definition of magnetic dipole moment vector 𝜇𝜇⃗ is analogous to electric dipole moment vector 𝑝𝑝⃗

Magnitude of magnetic dipole moment vector:

𝜇𝜇 = 𝑖𝑖𝑖𝑖

Direction of magnetic dipole moment is given by RH rule.

11. Spin Magnetic Dipole Moment Just as electrons have the intrinsic properties of mass and charge, they have an intrinsic property called spin. This means that electrons, by their very nature, possess these three attributes. You’re already comfortable with the notions of charge and mass. To understand spin it will be helpful to think of an electron as a rotating sphere or planet. However, this is no more than a helpful visual tool.

Imagine an electron as a soccer ball smeared with negative charge rotating about an axis. By the right hand rule, the angular momentum of the ball due to its rotation points down. But since its charge is negative, the spinning ball is like a little current loop flowing in the direction opposite its rotation, and the ball becomes an electromagnet with the N pole up. For an electron we would say its spin magnetic dipole moment vector, μ s , points up. Because of its spin, an electron is like a little bar magnet.

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Potential Energy of a Dipole

It is work required to turn dipole moment against the field �⃗ 𝑈𝑈 = −𝜇𝜇⃗. 𝐵𝐵