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Ampere’s Circuital Law states the relationship between an integrated magnetic field around a closed loop and the electric current passing through the loop. Ampere’s Law is derived from the Biot-Savart Law and provides an alternative approach for the calculation of the magnetic field caused due to a given current distribution. The main purpose of the Biot-Savart law was to calculate magnetic responses at molecular and atomic levels.
Ampere’s law can be defined as the magnetic field which is developed by an electric current proportional to the size of that electric current with a constant of proportionality that is equivalent to the permeability of free space. Ampere’s Circuital Law formula can be mathematically represented as: ∫B.dl = μo I
Where,
- μo = permeability of free space (i.e., 4 π × 10-15 N/ A2)
- ∫ B.dl = line integral of B surrounding a closed path
Read Also: Lenz’s Law
| Table of Content |
Key Terms: Ampere’s Circuital Law, Biot Savart Law, Magnetic Field, Ampere’s Loop, Electric Current, Magnetic Field, Electromagnetic Induction, Maxwell’s equation
What is Ampere's Law?
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Ampere’s Law states that the “magnetic field produced by an electric current is proportional to the size of the electric current with constant proportionality that is equivalent to the permeability of free space.”
Ampere’s circuital law states the relationship between an integrated magnetic field around a closed loop and the electric current passing through the loop. Ampere’s law is backed up by Maxwell’s equation which is:
| \(\bigtriangledown \times H = \frac{\delta D}{\delta t} + J\) |
Ampere’s Circuital Law Video Explanation
Ampere’s Circuital law is the alternative way to express Biot-Savart Law. Comparatively, Circuital law is nothing new when compared to the contents of Biot-Savart law.
- Both laws specify and link magnetic fields and currents. The physical significance of a steady electrical current is also expressed by both laws.
- Equipment that generates high magnetic fields also passes through a situation of high symmetry. In this situation, Ampere’s circuital law can be conveniently applied.
Ampere’s Circuit Law
Ampere’s Law Statement
Ampere’s Law states the association of magnetic fields with the electric current generated in them. According to Ampere's law formula, it is specified that the magnetic field is related to the given current or contrariwise on the condition that the electric field undergoes no modification with time.
Determine Magnetic Field Via Ampere’s LawQues. Assume that you have a long wire carrying a constant current I in amps. Determine the magnetic field which wraps the wire at any distance r from the wire. Ans. To determine the magnetic field wrapping the wire at any distance r, we first have to represent it:
Wire Carrying Current Diagram
Imaginary Loop |
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What is Ampere’s Circuital Law?
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Ampere's Circuital Law is the line integral of a magnetic field which surrounds a closed loop equivalent to the number of times the algebraic sum of currents passes via the loop.
According to Ampere’s law, the integrated magnetic field density (B) along an imaginary closed path equals the product of the current enclosed by the path and the permeability of the medium. Thus, Ampere Circuital Law Formula is as follows:
| \(\oint H.dl = I_{enc}\) |
Where,
- μ0 = Permeability of the medium
Considering that a conductor carries current I, then the current flow can yield a magnetic field enclosing the wire. It is assumed through the law that the closed-loop consists of small rudimentary parts of length (dl).
In the left side, the equation expresses that if an imaginary path surrounds the wire and the magnetic field has been added at every stage, then it can be said that it is numerically equivalent to the current surrounding this route (denoted by Ienc).
The total magnetic field density of the closed-loop is the integral of the magnetic field and the elemental length (Bdl).
This closed loop is also known as the Amperian Loop. The integral of the magnetic field is equal to the product of the net current passing through the closed-loop and the permeability of the medium (0 I). Thus,
\(\oint B.dl = \mu_0 I\)
This equation was derived by Maxwell. It alternatively explains that magnetic field intensity (H) through an imaginary closed path equals the current enclosed by the path.
⇒ Bdl = 0 I
⇒ B0dl = I
⇒ Hdl = I
\(\therefore\) H = B0
Ampere’s Circuital Law is influenced by a steady electrical current that does not modify or change with time.
Read More: Moving Charges and Magnetism Important Questions
Proof of Ampere’s Circuital Law
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Ampere's Circuital law can be demonstrated by:
Example 1: Regular Coil
Current carried by the regular coil = I
Small element on the loop = dl
∫ B.dl =∫ Bdl cos0 = B ∫ dl = small angle of the magnetic field.
As the magnetic field will be around a conductor we can assume the value of =0°
Magnetic field due to long current carrying wire at a perpendicular distance ‘r’ from the conductor is:
B=0 I/(2πr)
Due to the symmetry, the magnetic field does not vary. The whole circle of the circumference (2πr) is formed by the integral of an element:
∫ dl=2πr
After putting the value of B and ∫ dl:
B ∫ dl=0 I/(2πr)*2πr=0 I
∴∫ B.dl = 0 I
Example 2: Irregular Coil
An irregular coil means a randomly shaped coil. As the coil is irregular the radius ‘r’ will not remain constant.
∫ Bdl1= ∫ 0 I/(2πr)*dl1
We know, dθ1=dl1/r1
\(\therefore\) 0 I/(2πr1)*dl1=0 i/(2π) ∫ dθ1=0 I
⇒ ∫ B.dl=0 I
- The above derivations prove that the shape of the Amperian loop is not determined by the magnetic field.
- The magnetic field remains the same throughout the points of the Amperian Loop.
- Whether a coil is regular or irregular, Ampere's Circuital Law can be conveniently applied to both.
Read More: Synchrotron
What are the applications of Ampere’s Law?Ans. Some applications of Ampere’s Law are:
State ampere's circuital law.Ans. Ampere's law definition is “the line integral of a Magnetic field intensity moving along a closed path is equivalent to the current distribution passing via that loop.” |
What is Amperian Loop?
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Amperian loop is utilized by Ampere’s circuital law to point out the magnetic field in a particular region. Each point of the Amperian loop displays either of the following:
- B is a non-zero constant and is tangential to the loop.
- Or, B is normalized to the loop.
- Or, B simply disappears.
Here, the induced magnetic field = B.
The current inside the Amperian Loop determines only the line integral of the magnetic field.
Amperian loop
State true or false: Ampere’s law can be used to find the magnetic field inside a toroid.Ans. True. State true or false: Assuming that the direction of the current is reversed, the direction of magnetic field also reverses.Ans. True. |
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Previous Year Questions
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- In the shown arrangement of the experiment of the meter bridge … [JEE Advanced 2003]
- Which of the following set-up can be used to verify Ohm's law … [JEE Advanced 2003]
- 24 cells of emf 1.5V each having internal resistance of 11 ohm … [JKCET 2013]
- Figure shows three resistor configurations … [JEE Advanced 2008]
- A 100 W bulb B1 and two 60 W bulbs B2 and B3 … [JEE Advanced 2002]
- A meter bridge is set-up as shown in figure, to determine an unknown resistance … [JEE Advanced 2011]
- Find out the value of current through … [JEE Advanced 2005]
- A moving coil galvanometer of resistance … [JEE Advanced 2005]
- A rigid container with thermally insulated walls contains a coil of resistance … [JEE Advanced 2005]
- If now we hav.e to change the null point at 9th wire … [DUET 2007]
- The electrical permittivity and magnetic permeability of … [DUET 2003]
- Just after key K is pressed to complete the circuit, the …. [KEAM 1999]
- The resistance between any two terminals is when connected …. [NEET 1993]
Things to Remember
- Ampere’s Circuital Law states the relationship between an integrated magnetic field around a closed loop and the electric current passing through the loop.
- Ampere’s Law is derived from the Biot-Savart Law and provides an alternative approach for the calculation of the magnetic field caused due to a given current distribution.
- According to Ampere Circuital law, it is specified that the magnetic field that is related to the given current or contrariwise on the condition that the electric field undergoes no modification with time.
- Amperian loop is utilized by Ampere’s circuital law to point out the magnetic field in a particular region.
- According to Ampere’s law, the integrated magnetic field density (B) along an imaginary closed path equals the product of the current enclosed by the path and the permeability of the medium.
Read More:
Sample Questions
Ques. What rule do you need to follow if two parallel current-carrying wires in the same direction attract or repel? (2 marks)
Ans. If two wires are carrying current in the same direction, the right-hand thumb rule and Fleming’s left-hand rule can be applied to clear the confusion.
Right-hand thumb rule: Grab the wire with your right hand with your extended thumb pointing towards the direction of the current. You will notice your fingers swirling in the direction of the magnetic field.
Ques. Are Ampere’s law and Biot-Savart's law the same? (2 marks)
Ans. Biot-Savart law may have been derived from Ampere’s law or Ampere’s law may be derived from Biot-Savart law. Certain symmetrical conditions reveal that Ampere’s law is more useful.
Ampere's law is commonly based on mathematical results whereas Biot-Savart's law is based on experimental results.
Ques. Why do we only consider the current enclosed by the loop and not outside the loop? (2 marks)
Ans. Circuital refers to the enclosed loop hence, current outside the loop is not defined. Ampere’s law basically defines the magnetic field produced by electric current. The law only reveals the current of your interest produced by a magnetic field.
Ques. How is Ampere’s Circuital law significant? (2 marks)
Ans. The magnetic field of spherical, cylindrical and rectangular symmetries is obtained conveniently through Ampere’s law. The physical significance of the law can be understood after Maxwell’s corrections it presented the symmetry between electricity and magnetism.
Ques. Are there any limitations to this law? (2 marks)
Ans. Yes, there are certain limitations to Ampere’s circuital law. It is applicable only in magnetostatics, simply stating it is applicable only for steady currents.
Ques. If a lengthy empty pipe transports a direct electric current, will the force field related to the current be inside? (2 marks)
Ans. No, the magnetic field associated will be only outside the pipe. As the law states Bdl=0 a closed path inside a pipe will have no current enclosed making the Inside of the pipe zero. There would only be fields outside the pipe.
Ques. Derive the expression for the torque acting on a current-carrying loop placed in a magnetic field. (CBSE 2019) (3 marks)
Ans. Let's take the force on arm AB as f1 and the force on arm CD as f2.
F1 = F2 = IbB
The net torque is given by the sum:
T = I(a/2)bB + I(a/2)bB
T = IAB
Where A is considered as the area of the loop and the magnetic moment of the loop.
m→= IA→
Therefore,
T = mB
Hence, the expression for the torque acting on a current-carrying loop placed in a magnetic field.
Ques. A square loop of side ‘a’ carrying a current I2 is kept at a distance x from an infinitely long straight wire carrying a current I1 as shown in the figure. Obtain the expression for the resultant force acting on the loop. (CBSE 2019) (5 marks)
Ans. As said in the right-hand screw rule, the magnetic field can be seen across the loop into the plane with the force on length AD.
F = Bil
Force on length BC
F = Bil
Force on AB and CD will be equal and opposite, thus they will get cancelled. Force on the loop:
Ques. Two identical circular coils, P and Q each of radius R, carrying currents 1 A and 3 A respectively, are placed concentrically and perpendicular to each other lying in the XY and YZ planes. Find the magnitude and direction of the net magnetic field at the centre of the coils. (CBSE 2017) (5 marks)
Ans. √( There are two similar coils that are lying in perpendicular planes and have a common centre. P and Q carry current I and √3 I respectively.
Now, the magnetic field at the centre of P due to its current I is:
And magnetic field at the centre of Q, due to its current √3 I
Ques. Write the expression for the magnetic force acting on a charged particle moving with velocity ν in the presence of magnetic field B. (CBSE 2016) (2 marks)
Ans. F→ = q(v→ * B→)
F = qvB Sinθ and Force (F) is ⊥ to the plane that has v→ and B→
Ques. State Ampere's circuital law. Use this law to obtain the expression for the magnetic field inside an air-cored toroid of average radius 'r' having 'n' turns per unit length and carrying a steady current I. (5 marks)
An observer to the left of a solenoid of N turns each cross-section area 'A' and observes that a steady current I in it flows in the clockwise direction. Depict the magnetic field lines due to the solenoid specifying its polarity and show that it acts as a bar magnet of magnetic moment m = NIA. (2015)
Ans. Ampere’s circuital law states that the line fundamental to the induction of the magnetic field along a restricted curve is equivalent to the overall current that passes through the plane enclosed in the restricted curve times the permeability of the medium.
Application of Ampere’s Law for the following:
The spectator sees the south pole as given in the question paper and the magnetic moment because of the loop that is given by the following:
M = iA
For N turns, i = NI
M = NIA
Ques. Two long straight parallel conductors carry steady current I1 and I2 separated by a distance d. If the currents are flowing in the same direction, show how the magnetic field set up in one produces an attractive force on the other. Obtain the expression for this force. Hence define one ampere. (CBSE 2016) (5 marks)
Ans. When wire a is on b, the magnetic field formed is:
The force of current that is carried on a wire which is placed in the magnetic field = I(I→ * B→)
Therefore the force created on b :
The above expression shows the direction in which the force is experienced from wire a to b.
Similarly, we can say that the force is attractive by watching the direction of the force by the b wire that is being experienced towards the wire.
If we place two lengthy wires in a vacuum and separate them by 1m, one of the Ampere would be considered as the current in each wire that would be producing a force of wire in length per unit.
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