• in Magnetostatics, the Magnetic Field Is Produced by Steady Currents. The

Magnetostatics

• Magnetostatics is the branch of electromagnetics LECTURE 25: AMPERE’S LAW OF FORCE dealing with the effects of electric charges in AND BIOT-SAVART LAW steady motion (i.e, steady current or DC). • A fundamental law of magnetostatics is Ampere’s law of force which is based on physical observation and cannot be deduced logically or mathematically from any other physical law. • Ampere’s law of force is analogous to Coulomb’s law in electrostatics.

Magnetostatics (Cont’d) Ampere’s Law of Force

• In magnetostatics, the magnetic field is • Ampere’s law of force is the “law of action” produced by steady currents. The between current carrying circuits. magnetostatic field does not allow for • Ampere’s law gives the magnetic force between two current carrying circuits in an otherwise – inductive coupling between circuits empty universe. – coupling between electric and magnetic fields • Ampere’s law of force involves complete circuits since current must flow in closed loops.

1 Ampere’s Law of Force (Cont’d) Ampere’s Law of Force (Cont’d)

• Experimental facts: • Experimental facts:

– Two parallel wires F21 F12 – A short current- F12 = 0

carrying current in ×× carrying wire × I2 the same direction oriented I1 I2 I1 attract. perpendicular to a F long current-carrying – Two parallel wires F21 12 carrying current in ×• wire experiences no the opposite force. I1 I2 directions repel.

Ampere’s Law of Force (Cont’d) Ampere’s Law of Force (Cont’d) • The direction of the force established by the • Experimental facts: experimental facts can be mathematically represented by – The magnitude of the force is inversely proportional to the distance squared. unit vector in direction unit vector in direction of current I of current I1 – The magnitude of the force is proportional to 2 the product of the currents carried by the two aˆF = aˆ2 × (aˆ1 × aˆR ) wires. 12 12

unit vector in unit vector in direction direction of force on of I2 from I1 I2 due to I1 Copyright © 2002 by James T. Aberle LECTURE 25 Copyright © 2002 by James T. Aberle LECTURE 25 All rights reserved. All rights reserved.

2 Ampere’s Law of Force (Cont’d) Ampere’s Law of Force (Cont’d)

• The force acting on a current element I2 dl2 by a • The total force acting on a circuit C2 having a current element I1 dl1 is given by current I2 by a circuit C1 having current I1 is given by

µ I dl 2 × (I dl1 × aˆ ) µ 0 2 1 R12 dl 2 ×(dl1 × aˆ ) 0 I1I2 R12 F 12 = 2 F 12 = 2 4π R12 4π ∫∫ R CC21 12 Permeability of free space -7 µ0 = 4π×10 F/m

Ampere’s Law of Force (Cont’d) Magnetic Flux Density

• The force on C1 due to C2 is equal in • Ampere’s force law describes an “action at a magnitude but opposite in direction to the distance” analogous to Coulomb’s law. • In Coulomb’s law, it was useful to introduce the force on C2 due to C1. concept of an electric field to describe the interaction between the charges. F 21 = −F 12 • In Ampere’s law, we can define an appropriate field that may be regarded as the means by which currents exert force on each other.

3 Magnetic Flux Density (Cont’d) Magnetic Flux Density (Cont’d)

• The magnetic flux density can be introduced by •where writing µ I dl1 × aˆ 0 1 R12 µ (I dl1 × aˆ ) B12 = 2 0 1 R12 ∫ F 12 = I dl 2 × 4π C R12 ∫∫2 4π R2 1 CC2112 the magnetic flux density at the location of = I dl 2 × B12 dl due to the current I in C ∫ 2 2 1 1 C2

Magnetic Flux Density (Cont’d) Magnetic Flux Density (Cont’d)

• Suppose that an infinitesimal current element Idl • The total force exerted on a circuit C carrying is immersed in a region of magnetic flux density current I that is immersed in a magnetic flux B. The current element experiences a force dF density B is given by given by F = I dl × B d F = Idl × B ∫ C

4 Force on a Moving Charge Lorentz Force • If a point charge is moving in a region where both electric • A moving point charge placed in a magnetic field and magnetic fields exist, then it experiences a total force experiences a force given by given by

F m = qv× B Idl ⇔ qv F = F e + F m = q(E + v× B)

• The Lorentz force equation is useful for determining the The force experienced equation of motion for electrons in electromagnetic by the point charge is B deflection systems such as CRTs. q v in the direction into the paper.

The Biot-Savart Law The Biot-Savart Law (Cont’d)

• The Biot-Savart law gives us the B-field • The contribution to the B-field at a point P from a arising at a specified point P from a given differential current element Idl’ is given by current distribution. µ • It is a fundamental law of magnetostatics. 0 I dl′×(R − R′) d B(R) = 3 4π R − R′

5 The Biot-Savart Law (Cont’d) The Biot-Savart Law (Cont’d)

P • The total magnetic flux at the point P due to the R − R′ entire circuit C is given by Idl′ R µ I dl′×(R − R′) R′ B(R) = 0 ∫ 3 C 4π R − R′

The Biot-Savart Law (Cont’d)

• For a surface distribution of current, the B-S law becomes µ J s (R′)×(R − R′) B(R) = 0 ds′ ∫ 3 S′ 4π R − R′

• For a volume distribution of current, the B-S law becomes µ J (R′)×(R − R′) B(R) = 0 dv′ ∫ 3 V ′ 4π R − R′