Ampere’s Force Law
Ampere’s Force Law states that the force of attraction or repulsion between two wires carrying currents is proportional to their lengths and the intensities of current passing through them. If the currents flow in the same direction, repulsion takes place. If currents are flowing in opposite directions, attraction takes place. The law is based on these two basic concepts of electrostatics:
Biot-Savart Law states that every current carrying wire produces a magnetic field around it, as shown in the below figure.
Lorentz force refers to the force that every magnetic field exerts on any electric charge moving in its field.
Thumb Rule
1.8.1 Illustrative Problems
1. Six parallel aluminum wires of small, but finite, radius lie in the same plane. The wires are separated by equal distances d, and they carry equal currents I in the same direction. Let us determine the magnetic field at the center of the first wire. Assume that the currents in each wire is uniformly distributed over its cross section.
Solution:
Given:
Six parallel aluminum wires
Distance – d
Current – I
Formula to be used:
A schematic layout of the problem is shown in the below figure.
The magnetic field generated by a single wire is equal to
where r is the distance from the center of the wire.
The above equation is correct for all points outside the wire and can therefore be used to determine the magnetic field generated by wire 2, 3, 4, 5, and 6.
The field at the center of wire 1, due to the current flowing in wire 1, can be determined using Ampere's law and is equal to zero. The total magnetic field at the center of wire 1 can be found by vector addition of the contributions of each of the six wires. Since the direction of each of these contributions is the same, the total magnetic field at the center of wire 1 is equal to
2. A balance can be used to measure the strength of the magnetic field. Consider a loop of wire, carrying a precisely known current, shown in the figure below which is partially immersed in the magnetic field. The force that the magnetic field exerts on the loop can be measured with the balance and this permits the calculation of the strength of the magnetic field. Suppose that the short side of the loop measured 10.0 cm, the current in the wire is 0.225 A and the magnetic force is 5.35 x 10-2 N. Let us determine the strength of the magnetic field.
Current loop in immersed in magnetic field
Solution:
Given:
Side of the loop = 10.0 cm
Current in the wire = 0.225 A
Magnetic force = 5.35 x 10-2 N
Current loop in immersed in magnetic field
Formula to be used:
Consider the three segments of the current loop shown in the above figure which are immersed in the magnetic field. The magnetic force acting on segment 1 and 3 have equal magnitude, but are directed in an opposite direction and therefore cancel.
The magnitude of the magnetic force acting on segment 2 can be calculated as
This force is measured using a balance and is equal to 5.35 x 10-2 N. The strength of the magnetic field is thus equal to
2.1.1 Inconsistency of Ampere’s Law and Displacement Current Density
From Ampere’s law in magneto-statics, we have learned that
Taking the divergence of this equation gives
That is,
However, the continuity equation (conservation of charge) requires that
We can see that the above equations agree only when there is no time variation or no free charge density. Ampere’s law in magneto-statics is only valid for static fields and consequently, it violates the conservation of charge principle if we try to directly use it for time varying fields.
James Clerk Maxwell is the Father of Classical Electromagnetism. He combined the results of Coulomb's, Ampere's and Faraday's laws and added a new term to Ampere's law to form the set of fundamental equations of classical EM called Maxwell's equations.
This brings it into congruence with the conservation of charge law. Maxwell's contribution was to modify Ampere's law to read.
The term is called the displacement current density.
2.1.2 Maxwell’s Equations in Different Final Forms and Word Statements
Maxwell's Equations are commonly written in a few different ways. The form we have is known as point form.
The above equations are known as "point form" because each equality is true at every point in space. However, if we integrate the point form over a volume, we obtain the integral form. There is also Time-Harmonic Form and Maxwell's Equations written only with E and H. One form uses imaginary magnetic charge, which is useful for some problem solving.
Contained in the above equations is the equation of continuity,
In all the cases, the region of integration is assumed to be stationary.
WORD STATEMENT FORM OF FIELD EQUATIONS:
The word statements of the field equations may readily be obtained from the integral form of the Maxwell’s equations:
I. The mmf around a closed path is equal to the conduction current plus the time derivative of the electric displacement through any surface bounded by the path.
II. The emf around a closed path is equal to the time derivative of the magnetic displacement through any surface bounded by the path.
III. The total electric displacement through the surface enclosing a volume is equal to the total charge within the volume. IV. The net magnetic flux emerging through any closed surface is zero.
3.2 Uniform Plane Waves – Definition
"A uniform plane wave is an electromagnetic wave in which the electric and magnetic fields and the direction of propagation are mutually orthogonal, and their amplitudes and phases are constant over planes perpendicular to the direction of propagation."
For all the points in the plane which is perpendicular to the direction of propagation, the electric and magnetic fields are the same (or uniform).
For an electromagnetic UPW propagating in +z direction, we have
If the wave is propagating in + direction,we replace z by .
3.2.1 All Relations Between E & H
Plane waves in free space
For a plane wave in free space, we know that the E – field and H – field phasors to be
where,
Waves in a Dielectric Medium – Wave Equation
Waves in a Dielectric Medium – Dispersion Relation
Waves in a Dielectric Medium – Velocity
The velocity of waves in a dielectric medium is reduced from the velocity of waves in free space by the refractive index.
Velocity of waves in free space: c
Velocity of waves in dielectric medium of refractive index n: c/n
Waves in a Dielectric Medium – Wavelength
But the magnitude of the wave vector is related to the wavelength by the expression,
Thus for a dielectric medium, we get
Compare with for waves in free space.
The wavelength of plane waves in a dielectric medium is reduced from the wavelength of plane waves of same frequency in free space by its refractive index.
Waves in a Dielectric Medium – Magnetic Field
3.2.2 Sinusoidal Variations
In practice, most generators produce voltage and currents and hence electric and magnetic fields which vary sinusoidally with time. Further, any periodic variation can be represented as a weight sum of fundamental and harmonic frequencies.
Therefore, we can consider the fields having sinusoidal time variations, for example,
Here, = 2f = frequency of the variation.
Thus, every field or field component varies sinusoidally, mathematically by an additional term representing the sinusoidal variation. For example, the electric field can be represented as
Where is the time varying field.
The time varying electric field can be equivalently represented, in terms of corresponding phasor
quantity (r) as
The symbol ‘tilda’ placed above the E vector represents that is time – varying quantity
3.3 Wave Propagation in Loss less and Conducting Media
Plane Waves in Lossless Media
In a source free lossless medium,
where,
J = Current density
= Charge density
σ = Conductivity
Maxwell’s equations:
Take the curl of the first equation and make use of the second and the third equations, we have
This is termed as wave equation.
A similar equation for H can be obtained as
In free space, the wave equation for E is given by,
where,
c being the speed of light in free space (~ 3× 108 (m/s)). Hence the speed of light can be derived from Maxwell’s equation.
To simplify subsequent analyses, we consider a special case in which the field variation with time takes the form of a sinusoidal function:
sin ( ωt + φ) or cos ( ωt + φ)
Using complex notation, the E field can be written as,
Where is called the phasor form of E (x,y,z,t) and is in general, a complex number depends on the spatial co-ordinates only. The phasor form also includes the initial phase information and it is a complex number.
The benefits of using the phasor form are that,
Thus, differentiation or integration with respect to time can be replaced by multiplication or division of the phasor form with the factor jω. All other field functions and source functions can be expressed in the phasor form.
As all the time-harmonic functions involves the common factor ejωt in their phasor form expressions, we can eliminate this factor when dealing with the Maxwell’s equation.
The wave equation can now put in the phasor form as:
In phasor form, Maxwell’s equations can be written as,
Using the phasor form expression, the wave equation for E field is also termed as the Helmholtz’s equation, which is given by,
k is called as the wavenumber or the propagation constant.
where λ0 is the free space wavelength.
In an arbitrary medium with ε = ε0εr and μ = μ0μr,
We call,
This is termed as wavelength in the medium.
In Cartesian coordinates, the Helmholtz’s equation can be written as three scalar equations in terms of the respective x, y and z components of the E field.
For example, the scalar equation for the Ex component is given by,
Consider a special case of Ex in which there is no variation of Ex in the x and y directions, i.e.,
This is called the plane wave condition and Ex(z) now varies with z only. The wave equation for Ex becomes,
A plane wave is not physically realizable because it extends to an infinite extent in the x and y directions. However, when considered over a small plane area, its propagation characteristic is very close to a spherical wave, which is a real and common form of electromagnetic wave propagating.
Solutions to the plane wave equation take one form of the following functions, depending on the boundary conditions:
E0+ and E0- are constants to be determined by boundary conditions.
E0+e-jkz and E0-e+jkz are plane waves propagating along the +z direction and –z direction.