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**Straight line currents**

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**Straight line currents**

Φ1 Φ2 Loop Wire

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**…we have a bundle of straight wires and some of the wires passes through the loop…**

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**For surface current and volume currents:**

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**…for any arbitrary current distribution**

The Divergence and Curl of B …for any arbitrary current distribution rs P(x,y,z) dτ/(x/,y/,z/)

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**Applications of Ampere’s Law:**

…in Straight wire Sheet of Current Long Solenoid Toriodal Coil

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…Straight Wire Amperian Loop s I B

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**…Infinite Sheet of Current**

z K y z y x Amperian Loop x

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Long Solenoid K

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Problem 5.13 A steady current I flows down a long cylindrical wire of radius a. Find the magnetic field, both inside and outside the wire, if (a) The current is uniformly distributed over the outside surface of the wire. (b) The current is distributed in such a way that J is proportional to s, the distance from the axis. I a

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Problem: 5.15 Two long coaxial solenoids each carry current I, but in opposite directions as shown. The inner solenoid (radius a ) has n1 turns per unit length, and the outer one(radius b) has n2 turns per unit length. a b b

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Problem: A very long straight conductor has a circular cross-section of radius R and carries a current Iin. Inside the conductor there is a cylindrical hole of radius a whose axis is parallel to the axis of the conductor and a distance b from it. a R b Iin x y z

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**…find the magnetic field B at a point (a) on the x axis at x=2R and (b) on the y axis at y=2R.**

Iin x y z

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**…(c) at a point inside the hole.**

R b Iin x y z

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**Problem: 5.14 A thick slab extending from z=-a to z=+a carries a uniform volume current ,**

Find the magnetic field, as a function of z, both inside and outside the slab. x z y +a -a

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Problem:5.16 A large parallel plate capacitor with uniform surface charge density σ on the upper plate and –σ on the lower is moving with a constant speed v, as shown below. +σ -σ v

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Find, (a) The Magnetic field between the plates, (b) The Magnetic force per unit area on the upper plate & its direction, (c) …the speed v at which the magnetic force balances the electrical force. +σ -σ v

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**Toriodal Coil which along wire is wrapped.**

…a circular ring, or “donut” around which along wire is wrapped. ..the winding is uniform and tight enough so that each turn can be considered a closed loop.

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**Problem: …the magnetic flux through the end face of a solenoid…**

K

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**Comparison of Magnetostatics and Electrostatics**

The divergence and curl of the electrostatic field are: …together with the Boundary conditions determine the field uniquely.

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**Comparison of Magnetostatics and Electrostatics**

The divergence and curl of the Magnetostatics field are: …together with the Boundary conditions determine the field uniquely.

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**Magnetic Vector Potential**

…permits us to introduce a Vector Potential A in Magnetostatics:

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Problem: A spherical shell, of radius R, carrying a uniform surface charge σ, is set spinning at angular velocity ω. Find the vector potential A it produces at point P. ω r z R x y Φ/ Ө/ Ψ rs r/ da/ P ω R r P z

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**…Expressions for the Magnetic Field Inside & Outside the Spherical Shell are:**

(Inside the Spherical Shell)

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**…Expressions for the Magnetic Field Inside & Outside the Spherical Shell are:**

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Problem: 5.42 Calculate the Magnetic Force of Attraction between the Northern and Southern Hemispheres of a Spinning Charged Spherical Shell(…of Radius R carrying a uniform charge density σ and spinning at an angular velocity ω).

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…A Spinning Shell… ω K R (r=R at the surface)

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Problem: Find the vector potential of an infinite solenoid with n turns per unit length, radius R, and current I.

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Problem:5.22 Find the magnetic vector potential of a finite segment of straight wire, carrying a current I. z2 z1 o I z s rs

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**Problem: 5.24 If B is uniform, show that,**

works. Is this result unique, or there are other functions with same divergence and curl.

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y x A Ax Ay z r o Here, ‘r ’ is the projection of ‘vector-r’ on x-y plane.

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**Problem: 5.23 What current density would produce the vector potential,**

(where k is a constant), in cylindrical coordinates ?

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**Multipole Expansion of the Vector Potential**

rs O r/ dr/=dl/ I

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Problem: 5.60 (a) Work out the Multipole expansion for the vector potential for a volume current J. (b) Write down the Monopole potential and prove that it vanishes. (c) Write the corresponding dipole moment m.

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Problem: Find the magnetic dipole moment of the “bookend-shaped” loop as shown below. All sides have length w, and it carries a current I. z x y I w

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**The Magnetic Field of a Pure Dipole**

Ө Φ y z m r P x

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**Problem:5.34 Show that the magnetic field of a dipole can be written in coordinate free form as:**

Ө Φ y z m r P x

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Problem: 5.34 A circular loop of wire, with radius R, lies in the xy-plane, centered at the origin, and carries current I running counterclockwise as viewed from the positive z-axis. z R

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**(a) What is its magnetic dipole moment**

(a) What is its magnetic dipole moment? (b) What is the (approximate) magnetic field at points far from the origin? (c) Show that, for points on the z-axis, the answer is consistent with the exact field when z>>R.

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Problem:5.35 A phonograph record of radius R, carrying a uniform surface charge σ, is rotating at constant angular velocity ω. Find its dipole moment. z ω R y x

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**Fields due to “Pure” dipole & Physical dipole.**

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**Problem: 5.55 A Magnetic dipole**

is situated at the origin, in an otherwise uniform magnetic field Show that there exists a spherical surface, centered at the origin, through which no Magnetic field line passes.

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**Magnetostatic Boundary Conditions**

Babove┴ K Bbelow┴

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**Magnetostatic Boundary Conditions**

BIIabove K BIIbelow

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Thus: “The Perpendicular Component of the Magnetic Field is Continuous across a surface current.” Whereas “The Component of B that is parallel to the surface but perpendicular to the current is discontinuous by an amount μ0K.

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**These Results can be Summarized as:**

where, ‘n’ cap is the unit Vector perpendicular to the Surface pointing upwards.

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Problem : Show that the Vector Potential A is continuous, but its derivative inherits a discontinuity across any boundary.

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Problem: 5.24 (a)…find the vector potential a distance s from an infinite straight wire carrying a current I. (b) …find the vector potential inside the wire if it has a radius R and the current is uniformly distributed.

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R z s J

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**END OF THE MAGNETOSTATICS**

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22.7 Source of magnetic field due to current

22.7 Source of magnetic field due to current

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