49 views
in Ampere’s Law by
What is Ampere's law equation?

Your answer

Thanks for your contribution. Feel free to answer this question. Please avoid short answer. Your answer is most welcome. Be genuine.

Upload image or document:

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
Are you a robot ? (Y = Yes / N = No)
To avoid this verification in future, please log in or register.

2 Answers

0 votes
by
 
Best answer

image image image image image image image image image

Ampere's law

NPTEL

Distributed under Creative Commons Attribution-ShareAlike - CC BY-SA.

0 votes
by

Module 3 : MAGNETIC FIELD Lecture 16 : Ampere's Law Objectives In this lecture you will learn the following Establish Ampere' law in integral form. Calculate the magnetic field for certain current configuration using Ampere's law. Derive the differential form of Ampere's law. Ampere's Law Biot-Savart's law for magnetic field due to a current element is difficult to visualize physically as such elements cannot be isolated from the circuit which they are part of. Andre Ampere formulated a law based on Oersted's as well as his own experimental studies. Ampere's law states that `` the line integral of magnetic field around any closed path equals times the current which threads the surface bounded by such closed path. . Mathematically, In spite of its apparent simplicity, Ampere's law can be used to calculate magnetic field of a current distribution in cases where a lot of information exists on the behaviour of . The field must have enough symmetry in space so as to enable us to express the left hand side of (1) in a functional form. The simplest application of Ampere' s law consists of applying the law to the case of an infinitely long straight and thin wire. Example 12 By symmetry of the problem we know that the magnitude of the field at a point can depend only on the distance of the point from the wire. Further, the field is tangential to the circle of radius , its direction being given by the right hand rule. Thus the integral around the circle is Equating this to , we get which is consistent with the result obtained from Biot-Savart's law. Let us see if the result above is consistent with a path which is not circular, as shown in the figure. The field at every element of the path is perpendicular to . From geometry, it can be seen that Thus We need to specify the direction along which the path is traversed. This is done by Right Hand Rule. If we curl the fingers of our right hand along the path of integration, the direction along which the thumb points is the direction of current flow. For the case where the path of integration lies totally outside the path of the current, for every element at P, there exists another element at P' for which has opposite sign. Thus when complete line integral is taken, the contributions from such pairs add to zero Combining these, we get Ampere's law in the form of Eqn. (1) Example 13 Calculate the field due to a uniform current distribution in an infinite wire of cross sectional radius . Solution : Let the cross section of the wire be circular with a radius . Take the current direction to be perpendicular to the page and coming out of it. Symmetry of the problem demands that the magnitude of the field at a point is dependent only on the distance of the point from the axis of the wire. Consider an amperian loop of radius . As before we have If (as in loop 1), the entire current is enclosed by the loop. Hence so that If (loop 2), the current enclosed is proportional to the area, i.e. so that The field distribution with distance is as shown. Exercise 1 A long wire of cross sectional radius carries a current . The current density varies as the square of the distance from the axis of the wire. Find the magnetic field for and for . ( Hint : First show that the current density and obtain an expression for current enclosed for . Answer : for and for .) Exercise 2 A hollow cylindrical conductor of infinite length carries uniformly distributed current I from . Determine magnetic field for all . (Answer : Field is zero for , for and for .) Exercise 3 A coaxial cable consists of a solid conductor of radius with a concentric shell of inner radius and outer radius . The space between the solid conductor and the shell is supported by an insulating material. A current goes into the inner conductor and is returned by the outer shell. Assume the cuurent densities to be uniform both in the shell and in the inner conductor. Calculate magnetic field everywhere. (Ans. inside the inner conductor, between the shell and the inner conductor, ) Exercise 4 Determine the magnetic field in a cylindrical hole of radius inside a cylindrical conductor of radius . The cylinders are of infinite length and their axes are parallel, being separated by a distance . The conductor carries a current of uniform density. (Hint : The problem is conveniently solved by imagining currents of equal and opposite densities flowing in the hole and using superposition principle to calculate the field. Answer : The field inside the hole is constant ) Example 14 We take the solenoid to be closely wound so that each turn can be considered to be circular. We can prove that the field due to such a solenoid is entirely confined to its interior, i.e. the field outside is zero, To see this consider a rectangular amperian loop parallel to the axis of the solenoid. Field everywhere on AB is constant and is . Likewise the field everywhere on CD is . By Right hand rule, the field on AB is directed along the loop while that on CD is oppositely directed. On the sides AD and BC, the magnetic field direction is perpendicular to the length element and hence is zero everywhere on these two sides. Thus By Ampere's law, the value of the integral is zero as no current is enclosed by the loop. Thus . The field outside the solenoid is, therefore, independent of the distance from the axis of the solenoid. However, from physical point of view, we expect the field to vanish at large distances. Thus . To find the field inside, take an amperian loop EFGH with its length parallel to the axis as before, but with one of the sides inside the solenoid while the other is outside. The only contribution to comes from the side GH. Thus, where is the current through each turn and is number of turns per unit length. because the number of turns threading the loop is . Hence, is independent of the distance from the axis. Exercise 5 A toroid is essentially a hollow tube bent in the form of a circle. Current carrying coils are wound over it. Use an amperian loop shown in the figure to show that the field within the toroid is , where is the number of turns and the circumference of the circular path. Note that as the circumference of the circular path varies with the distance of the amperian loop from the toroid axis, the magnetic field in the toroid varies over its cross section. Take the inner radius of the toroid to be 20cm and the outer radius as 21cm. Find the percentage variation of the field over the cross section of the toroid. (Ans. 2.9%) Example 15 An infinite conducting sheet carries a current such that the current density is per unit length. Take an amperian loop as shown. The contribution to the line integral of from the sides QR and SP are zero as is perpendicular to . For PQ and RS the direction of is parallel to the path. Hence giving . Exercise 6 Calculate the force per unit area between two parallel infinite current sheets with current densities and in the same direction. ( Ans. ) Ampere's Law in Differential Form We may express Ampere's law in a differential form by use of Stoke's theorem, according to which the line integral of a vector field is equal to the surface integral of the curl of the field, The surface is any surface whose boundary is the closed path of integration of the line integral. In terms of the current density , we have, where is the total current through the surface . Thus, Ampere's law is equivalent to which gives You may recall that in the case of electric field, we had shown that the divergence of the field to be given by . In the case of magnetic field there are no free sources (monopoles). As a result the divergence of the magnetic field is zero The integral form of above is obtained by application of the divergence theorem Thus the flux of the magnetic field through a closed surface is zero. Recap In this lecture you have learnt the following Ampere's law was stated in integral form and used to calculate magnetic field in symmetric situations. Calculation of magnetic field was done due to a a long straight wire, a coaxial cable, a solenoid, a toroid and a current sheet done using Ampere's law. Using Stoke's theorem Ampere's law can be expressed in a differential form.


NPTEL

Distributed under Creative Commons Attribution-ShareAlike - CC BY-SA.


Related questions

0 answers 119 views

What is the order of the characteristic equation of an armature controlled DC motor?  (A) First order equation (B) Second order equation (C) Zero order equation (D) Third order equation 

asked Apr 5, 2018 in Electrical Engineering by Shimroz123
1 answer 41 views

What is the EMF equation of the transformer?

asked Dec 22, 2017 in Fundamentals of electrical power systems by Quiz
2 answers 19 views

What is meant by Biot Savart law?

asked May 30, 2018 in Biot-Savart’s laws by Shimroz123
2 answers 78 views

What is the Lenz's law?

asked Mar 14, 2018 in Electrical Engineering by Quiz
1 answer 29 views

What is KCL and KVL law?

asked Mar 10, 2018 in Electrical Engineering by Quiz
1 answer 20 views

What is the formula for Watts Law?

asked Jan 4, 2018 in Electrical Engineering by Zeeshan
2 answers 44 views

What is Coulomb's law ?

asked Nov 11, 2017 in Power Electronics by Quiz
1 answer 47 views

For a given base voltage and base volt-amperes, the per unit impedance value is x. For the doubled base values of both voltage and volt-amperes, the per unit impedance will be (a) 2x (b) 0.25x (c) 0.5x (d) no change 

asked May 25, 2018 in Battery by Shimroz123
1 answer 42 views

A 50 Hz, 4-pole, 500 MVA, 22 kV turbo-generator is delivering rated megavolt amperes at 0.8 power factor. Suddenly a fault occurs reducing in electric power output by 40%. Neglect losses and assume constant power input to the shaft. The accelerating torque in the generator in MNm at the time of fault will be  (A) 1.528 (B) 1.018 (C) 0.848 (D) 0.509

asked May 25, 2018 in Power system by Shimroz123
1 answer 6 views

The number of micro amperes in 2 milli amperes is? A.2 μA B.20 μA C.200 μA D.2,000 μA

asked May 15, 2018 in Basic concepts by Shimroz123
1 answer 13 views

The ratio of full -load volt-amperes to short-circuit volt-amperes is equal to :

asked Apr 30, 2018 in Electrical Engineering by Shimroz123
0 answers 18 views

If the transmission line is loaded with leading reactive volt amperes, there will be rise of voltage at the receiving end and this (a) increases (b) decreases (c) remains same with the increase in the length of line (d) none of these

asked Apr 14, 2018 in Electrical Engineering by Shimroz123
1 answer 23 views

A current of 4 amperes flowing for 45 minutes deposits 1.062 gm of metal at cathode. Calculate the equivalent weight of the metal. (Given 1 Faraday = 96500 C)

asked Sep 3, 2018 by Shimroz123
1 answer 18 views

Four amperes of current are measured through a 24 ohms resistor connected across a voltage source. How much voltage does the source produce? A.960 V B.9.6 V C.96 V D.8 V

asked May 15, 2018 in Basic concepts by Shimroz123
0 answers 33 views

The characteristic equation of a simple servo system is s2+6s+25=0. Damping factor of the system is: A) 3.2 B) 2.4 C) 1.8 D) 2.9

asked Jun 17, 2018 in Electrical Engineering by Shimroz123
1 answer 11 views

The angle δ in the swing equation of a synchronous generator is the  (A) angle between stator voltage and current (B) angular displacement of the rotor with respect to the stator (C) angular displacement of the stator mmf with respect to a synchronously rotating axis. (D) angular displacement of an axis fixed to the rotor with respect to a synchronously rotating axis

asked May 17, 2018 in Synchronous machine by Shimroz123
1 answer 23 views

The characteristic equation for a closed loop system with forward gain K is s4 + 4s3 + 8s2 + 6s + K = 0. The critical gain value Kc for stability should not exceed  (a) 3.25 (b) 9.75 (c) 13.0 (d) 23.3

asked May 15, 2018 in Control System by Shimroz123
1 answer 6 views

The swing equation for a synchronous machine is based on (A) relative motion between load angle and stator magnetic field (B) maximum power flow possible through a particular point (C) net torque and angular displacement (D) relation between excitation voltage and excitation current

asked Apr 25, 2018 in Electrical Engineering by Shimroz123
1 answer 17 views

A discrete time system is stable if all the roots of the characteristic equation. lie (a) Outside the circle of unit radius (b) Within the circle of unit radius (c) Outside the circle of radius equal to 3 -units (d) On the circle of infinite radius

asked Apr 23, 2018 in Control System by Shimroz123
0 answers 27 views

In the case of second order differential equation if the damping ratio is unity, then the poles are  A) Equal, positive and real B) Imaginary and complex conjugate C) Equal, negative and real D) None of the above

asked Apr 9, 2018 in High voltage engineering by Shimroz123
0 answers 30 views

In the solution of load flow equation, Newton Raphson (NR) method is superior to the GaussSeidal (GS) Method, because the:  (A) Convergence characteristic of the NR methods are not affected by selection of slack bus (B) Number of iterations required in the NR method is not independent of the size of the system (C) Time ... the NR method is less when compared to the GS method (D) Number of iteration required in the NR method is more than compared to that in the GS method 

asked Apr 5, 2018 in Electrical Engineering by Shimroz123
1 answer 37 views

If the characteristics equation of a closed loop system is s2+2s+2=0, then the system is  (A) Over damped (B) Critically damped (C) Under damped (D) Undamped 

asked Apr 5, 2018 in Electrical Engineering by Shimroz123
0 answers 18 views

Which among the following is the equation for the number of independent loops for a network with n nodes and b branches?  (A) n-1 (B) b+n-1 (C) b-n+1 (D) b-n

asked Apr 5, 2018 in Electrical Engineering by Shimroz123
0 answers 15 views

The equation V·j = 0 is known as : (A) Poissons’s equation (B) Laplace equation (C) Continuity equation (D) Maxwell equation

asked Mar 31, 2018 in Electrical Engineering by Shimroz123
2 answers 239 views

The equation of 50 Hz current sine wave having r.m.s. value of 60 A is?

asked Mar 25, 2018 in Electrical Engineering by Shimroz123
1 answer 44 views

Which law is synonymous to the occurrence of diamagnetism?   (a) Ampere’s law (b) Maxwell’s law (c) Coulomb’s law (d) Lenz’s law 

asked May 13, 2018 in Electrical Engineering by Shimroz123
1 answer 17 views

Kirchhoff's current law is applicable to 1. Closed loops in a circuit 2. Junction in a circuit 3. Magnetic circuits Which of the above is/are correct ?  (a) 1 only (b) 2 only (c) 3 only (d) 1, 2 and 3

asked Apr 25, 2018 in Electrical Engineering by Shimroz123
1 answer 9 views

The Kirchoff's law for a.c. is applicable upon consideration of -  a) Phasor sum b) Algebraic sum c) Progression sum d) Linear sum

asked Apr 10, 2018 in Instrumentation and control by Shimroz123
0 answers 31 views

Ohm’s law is applicable to : (A) Semi-conductors (B) Vacuum tubes (C) Electrolytes (D) Semiconductors and Vacuum tubes

asked Mar 31, 2018 in Electrical Engineering by Shimroz123
1 answer 15 views
1 answer 10 views

Kirchhoff s law is not applicable to circuits with?

asked Mar 25, 2018 in Network Analysis by Shimroz123
1 answer 19 views

Kirchhoff s law is applicable to?

asked Mar 25, 2018 in Network Analysis by Shimroz123
0 answers 10 views

Kirchhoffs voltage law is related to?

asked Mar 25, 2018 in Network Analysis by Shimroz123
1 answer 22 views

Kirchhoffs current law is applicable to only?

asked Mar 25, 2018 in Network Analysis by Shimroz123
1 answer 34 views

The condition for the validity under Ohm's law is that?

asked Mar 25, 2018 in Basic concepts by Shimroz123
2 answers 70 views

why ohm's law is not applicable for ac circuit?

asked Oct 2, 2017 in Electrical Engineering by Quiz
1 answer 36 views
1 answer 65 views

is Ohms law applicable to vacuum tubes?

asked Aug 16, 2017 in Electrical Engineering by Zeeshan
0 answers 30 views
asked Oct 15, 2018 by anonymous
0 answers 35 views
asked Apr 10, 2018 in Electrical Engineering by Narendra
0 answers 144 views

The system described by the characteristic equation S4 +2S3 +3S2 +4s+5=0 has  A) No roots have positive real part B) One root has positive real part C) Two roots have the positive real parts D) None of these

asked Apr 9, 2018 in Electronics Engineering by Shimroz123
0 answers 66 views

In a network containing resistances and reactances the roots of the characteristic equation give for the circuit : (A) The forced response (B) The total response (C) The natural response (D) The damped response

asked Mar 31, 2018 in Electrical Engineering by Shimroz123

Welcome to electronics2electrical.com here you can ask questions related to electrical, electronics, mechanical, telecommunication, instrumentation, computer, mathematics, physics etc.
Be respectful to all the members. Do not copy and paste the answers from other websites which have copyright content. While asking question give full details about it.

Categories

Most popular tags

power motor dc circuit transformer voltage current used system phase resistance factor synchronous load ac energy induction electric generator series frequency between speed capacitor use electrical meter line difference control type mosfet transmission magnetic plant high single instrument bjt unit source advantages function diode and machine winding field define torque parallel amplifier supply shunt thyristor motors electricity arduino maximum time relay armature problem value on transformers types coil diagram state flow ratio material three formula starting direction theorem method emf operating efficiency digital wave microprocessor test instruments inductance loss measure operation connected signal low applications effect single-phase network temperature working constant losses different law wattmeter measuring compare controlled breaker drive device logic rc full switch flux wire resistivity disadvantages free of materials machines angle force converter conductor transistor gain open protection scr core measurement number bridge principle generators reactance circuits negative the friction iron loop short pole battery conservation steam resistors hysteresis computer using analog lines secondary station gate a rectifier inverter linear induced relays nuclear capacitance basic characteristics design direct work rotor electronics ammeter forces diesel damping rlc connection factors capacitors minimum insulation moving regulation running self systems air fault range main stability quality starter igbt eddy alternator ideal rl average 3-phase plants arc thermal error fuzzy biasing dielectric pressure balanced superposition errors copper rotation feedback impedance measured electronic electrons charge inductive transfer explain start off back curve over solar is three-phase tariff locomotive peak bias zener engineering commutator surge conductors rating universal potentiometer density permanent mechanical transducer capacity memory adc excitation two fuse pure harmonics application semiconductor inductor internal pmmc reaction welding resonance traction permeability breakers rms designed electromagnetic si generation brushes switching capacitive shaded rate 1 distribution resistor methods delta star oscillator reluctance simplification algebra 8085 boolean weston dynamometer insulating strength installation definition fuel heating earth units

8,471 questions

7,069 answers

134 comments

3,088 users

...