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# Ampere's Law

Ampere’s law related the integrated magnetic field around a closed loop to the electric current passing through the loop. It was discovered in 1826 by Andrew-Marie Ampere . In was derived using hydrodynamics in 1861 by James Clerk Maxwell. Ampere’s law is now one of the Maxwell equations which form the basis of classical electromagnetism.

Using Ampere’s law, the magnetic field associated with a given current or current associated with a given magnetic field can be determined. This is provided that there is no time changing electric field presents. Ampere’s law can be written in two forms, integral and differential. These two forms are equivalent and can be related by the Kelvin-Stokes theorem. It can also be written in terms of B or H magnetic fields.

Simple electric current situations are classified when the electric current is “free current”. This is when the current passes through a wire or battery. In contrast, “bound current” is when bulk material can be magnetized and/or polarized.

All current is fundamentally the same. However, there are often practical reasons for wanting to treat bound current differently from free current. For instances, the bound current typically originates over atomic dimensions and one may wish to take advantage of a simpler theory intended for larger dimensions.

 Richard Fitzpatrick (2007). "Ampere's Circuital Law".

### Modified Ampere's Law Explanation

Show why the math for amperes time dependent equation is right. At a site they derive the amperes law for a changing electrical field. I know that B is related to current. But what I wanted was an illustration for how the mathematical justification in the attachment is right by pointing out where the dE/dt effects the magneti

### Maxwell's Equation for Static Electric and Magnetic Fields

2. (a) (i) Write down Maxwell's equations for static electric and magnetic ﬁelds in the vacuum (note that you should include charge and current densities). (ii) How did Maxwell modify Ampére's law to account for dynamic electric ﬁelds? (b) In a region of space in which the relative permittivity is (attached equation),

### Time Harmonic Form of Maxwell's Equations

Assume that both the E field an the B field are time-harmonic, so that each can be written as: E = E(0)exp(i(k*r-wt)) B = B(0)exp(i(k*r-wt)) where * = dot The time and spatial derivatives can then be written as partial of E with respect to t = -iwE Partial of B with respect to t = -iwB divergence of E = ik*E

### Single Phase Closed-Circuit

A closed-circuit, 500-turn coil, of resistance 100, and negligible inductance, is wound on a square frame of 40 cm side. The frame is pivoted at the mid points of two opposite sides and is rotated at 250 r/min in a uniform magnetic field of 60 mT. The field direction is at right angles to the axis of rotation. For the instan

### Wave solutions of Maxwell's equations and Ampere-Maxwell law

An electromagnetic signal is generated by a Hertzian dipole located at a point P, which has the position vector r = ?(100m) ez. The signal is detected by a small wire loop located at the origin. Apart from the dipole and the loop, the nearby space is empty. Experimentation reveals that the detected signal is induced by a cha

### Ampheres law symmetry

A long metal cylinder of radius (a) has the z-axis as its axis of symmetry. The cylinder carries a steady current of uniform current density (Vector)J=(J subscript (z)) ((vector)e subscript(z)). Derive an expression for the magnetic field at distance r from the axis where r<a. By resolving the cylindrical unit (vector)e subsc

### Finding Electric Field of Two Long Co-Axial Cylindrical Surfaces

Two infinitely long coaxial cylindrical surfaces have the z axis as their common axis. The inner surface of radius a carries a surface current K1 = k1 phi, and the outer surface of radius b carries a surface current K2 = k2 phi. Both k1 and k2 are constant. Find B everywhere. Please refer to the attachment for question with p

### Find the Magnetic Field H(r) in All Space

A loop with a current is located in the plane separating two media with magnetic permeability mu1 and mu2. Find the magnetic field H(r) in all space assuming that the magnetic field generated by the same loop with the same current in vacuum is known to an equal to H0(r). Answer is H1=2*mu1*H0(r)/(mu1+mu2), H2= 2*mu1*H0(r)/(mu

### Magnetization Dipole Force

A uniform current density J =Jo *zhat fills a slab straddling the yz plane, from x=-a to x=+a. A magnetic dipole m= mo *xhat is situated at the origin. a) Find the force on the dipole, using F = ▽(m.B) b) Do the same for a dipole pointing in the y direction m= mo *yhat

### The Magnitude and Direction of Magnetic Fields

Point P is midway between two long, straight, parallel wires that run north-south in a horizontal plane. The distance between the wires is 1cm. Each wire carries a current of 1A toward the north. A) Find the magnitude and direction of the magnetic field at point P. B) Repeat the question if the current in the wire on the e

### Magnetic field due to infinitely long cylinder

An infinitely long cylinder of circular cross section of radius a carries over current I^1 that is uniformly distributed over the cross-sectional area. The axis of the cylinder coincides with the z axis and I^1 is in the positive z direction. Choose a field point on the x axis and find B for all values of x, both inside and outs

### Electromagnetic Induction: Toroidal solenoid

A toroidal solenoid has a rectangular cross-section, with inner and outer radii a and b, and a height h. It has N uniformly spaced turns, with air inside. What is the magnetic flux through a cross-section of the toroid? What is the inductance of the toroidal solenoid?

### Electromagnetic Induction: Understanding Ampere's law

4) Investigate the following statement using Ampere's law: In a region of space where there are no conduction or displacement currents, it is impossible to have a uniform magnetic field that abruptly drops to zero.

### Magnetic Field of Coaxial Cable

A solid conductor with radius a is supported by insulating disks on the axis of a conducting tube with inner radius b and outer radius c. The central conductor and tube carry equal currents I in opposite directions. The currents are uniformly distributed over the cross sections of each conductor. This is known as a coaxial cable

### Electromagnetic Induction: Self and Mutual Inductance

An inductor consists of 500 turns of wire of resistance 6.0 &#937; wound tightly and uniformly on a toroidal ring of an insulating magnetic material with relative permeability &#956; = 40. The material is linear so &#956; is independent of the magnetic field. The mean radius of the toroid is 15 cm and the cross sectional area is

### Magnetic Field of a Current Carrying Wire

Find the magnetic field at a distance r from the center of a long wire that has radius 3.13 mm and carries a uniform current per unit area 208 A/m^2 in the positive z direction. A.) First find the magnetic field, B out (r) , outside the wire (i.e., when the distance r is greater than ) at a point 6.08 mm , from the center o

### Electromagnetism: C and L of a Parallel Wire Line

I need to find the capacitance and inductance per unit length between 2 parallel wires. To do this, I have to calculate the E and B fields between the 2 wires, each with radius a, separated by d, where d is much bigger than a? Could you please show the whole derivation? (I assume you'd use Gauss' and Amperes' law but I could nev

### To find the value and direction of the Poynting vector at the surface of a current carrying wire and show that the total power entering the wire per unit length is (I^2)R.

A straight conducting wire of circular cross section, radius a, has a resistance R per unit length and carries a constant current I. Find the value and direction of the Poynting vector at the surface of the wire, and hence show that the total power entering the wire per unit length is (I^2)R. By considering the divergence of the

### Lorentz Law of Force

See attached file for full problem description. Two infinitely long wires are placed parallel to each other at a distance d. One wire carries the current I_1 and the other carries the current I_2. Using the Lorentz Law of force, F = q (E + V x B), find the force density (i.e., force per unit length) that one wire exerts on th

### Magnetism: Field Flux and Ampere's Law

See attached file for full problem description. Questions - 10, 18, & 37.

### Magnetic Fields

Use Ampere's Law to find the magnetic field, produced by a lightning bolt at a distance of 37.0 m from the lightning bolt, which can be approximated by a long straight wire, when a charge of 10.0 C flows in a time of 6 x 10^-3 seconds.

### Magnetic Fields for a Long Solenoid

A long solenoid has 1400 turns per meter of length, and it carries a current of 3.5 A. A small circular coil of wire is placed inside the solenoid with the normal to the coil oriented at an angle of 90 degrees with respect to the axis of the solenoid. he coil consists of 50 turns, has an area of 1.2 x 10^-3 m^2, and carries a cu

### Ampere's Law: Magnetic Field Due to a Coaxial Cable

A coaxial cable consists of a solid inner conductor of radius R1, surrounded by a concentric cylindrical tube of inner radius R2 and outer radius R3. The conductors carry equal and opposite currents I distributed uniformly across their cross-sections. Determine the magnetic field at a distance R from the axis for: a) R < R1,

### Two parallel wires, each carrying a current of I = 3.1 A, are shown below, where d = 5.2 cm. The current in wire 1 is in the opposite direction of wire 2. Find the direction and magnitude of the net magnetic field at points A, B, and C.

1. Two parallel wires, each carrying a current of I = 3.1 A, are shown below, where d = 5.2 cm. The current in wire 1 is in the opposite direction of wire 2. Find the direction and magnitude of the net magnetic field at points A, B, and C. See attached file for full problem description. I don't know how to do the problem

### Magnitude and direction of magnetic field

"Two parallel wires, each carrying a current of I = 1.8 A, are shown in the figure below, where d = 7.1 cm. The current in wire 1 is in the opposite direction of wire 2. Find the direction and magnitude of the net magnetic field at points A, B, and C." Please see the attached for the full question.

### Magnetic field from two infinitely long wires

Two wires of infinite length are placed at a distance D = 10cm on opposite sides of a point O, both wires carry the same current of 1.0 A, in which direction must this current flow in order for the magnetic field strength at O to be non-zero. Consider a general point on the line joining the 2 wires and a distance d away from

### Magnitude of the magnetic field

Question attached regarding the magnitude of the magnetic field. #44

### Parallel Conductors

Question: Two long, parallel conductors, separated by 10.0 cm, carry currents in the same direction. The first wire carries current I1 = 5.00 A and the second carries I2 = 8.00 A. (a) What is the magnitude of the magnetic field created by I1 at the location of I2? (b) What is the force per unit length exerted by I1 on I2?

### Find the magnetic field, as a function of z, both inside and outside the slab.

See attached file for graphs. A thick slab extending from z=-a to z=+a carries a uniform volume current J=Jx (fig 5.41 attached). Find the magnetic field, as a function of z, both inside and outside the slab.

### Magnetic Field: Ampere's Law, magnetic field inside a solenoid

A. Use Ampere's Law to find the general equation for the magnetic field in an infinitely long solenoid. b. Find the magnetic field inside the solenoid.