Application of Ampere's Law - Magnetic Field of an Infinite Sheet of Current

Magnetic Field of an Infinite Sheet of Current:

We can consider an instance of a current distribution which does not have cylindrical symmetry, but which is susceptible to Ampere's Law for finding the magnitude of the magnetic field. Consider a sheet of current which is infinitesimally thin but infinitely long and wide. The sheet has a linear current density (i.e. current per unit length).

An infinite sheet of current with current per unit length. We wish to find the magnetic field direction and magnitude at point P a distance h away from the sheet.

Consider a set of wires laid in place of the current sheet. Each wire carries current out of the page. The magnetic field due to wires which are equidistant from the point directly underneath point P can only be in the horizontal direction.

We can replace the sheet with an infinite set of wires arranged so that each wire carries current consistent with an overall current per unit length of as in the case of the current sheet. We see in figure that wires which are equidistant from the line from the set of wires to point P have magnetic fields whose vertical components cancel and whose horizontal components add. Hence, the net magnetic field is in the horizontal direction. This field is uniform since the distribution of wires is infinite, i.e. any position for point P can be considered as the "middle" of an infinitely long current sheet. Note also that the magnetic field direction for the infinite sheet is independent of the distance of P from the sheet and that the same arguments for extending consideration of a finite set of wires to an infinite current sheet state that the magnetic field direction on the other side of the sheet has the field pointing in the opposite direction.

Application of Ampere's Law - Magnetic Field of a Toroid

Magnetic Field of a Toroid:

We can use Ampere’s Law to examine the case of another device that produces a uniform magnetic field. In this case, we take advantage of our analysis of the solenoid to ask what happens if we bend a solenoid into a circle so that the ends join. The new configuration still has approximately zero field in the regions outside the volume contained by the coils, but the field inside that volume is again approximately uniform if the distance between the coils is small compared to the size of the coils. This device is a toroid.

Ampere's Law applied to a toroid. Note that Amperian loops which lie wholly outside the volume contained by the toroidal coils experience no magnetic field.

We expect the magnetic field to have circular symmetry about the center of the toroid using the same reasoning as for the solenoid. Hence, we expect it to be most useful to use circular paths for evaluating the Ampere integral of magnetic field and path. For any circular path whose area is not intersected by the coils, the magnetic field is zero and the current penetrating the area is, by definition, zero.

Application of Ampere's Law - Magnetic Field of a Solenoid

Magnetic Field of a Solenoid:

We can readily find examples where Ampere's Law is more useful, by virtue of being easier to apply, than the Biot-Savart Law. For example, as was the case for the electric field, it was deemed to be very practical to have a device which could "store" magnetic field as a capacitor "stores" an electric field. We know that what the capacitor really stores is electric charge separation, but one of the properties for which a parallel-plate capacitor is useful is its ability to produce a uniform electric field between its plates. The equivalent device for magnetic fields is the solenoid. It's simply a conducting wire wrapped into a cylindrical shape. Even though the actual shape of the wire is helical, for densely packed wrapping we can actually consider the solenoid to be a bunch of closely spaced coils. Near the center of each coil we know that the magnetic field is nearly perpendicular to the plane of the coil.

The magnetic field near the center of a single coil carrying a current I.

The magnetic field near the center of a set of three coils all carrying a current I.

If we place coils on either side of the first and let them all carry the same current I in the same direction, then the magnetic field lines will link together near the center to make a field that is approximately uniform in direction and strength near the center of the set of coils. Placing many coils in close proximity to each other yields the solenoid field.

A solenoid approximates many current-carrying coils spaced much more closely than the coil diameter.

A perfect solenoid has coils so close together that the magnetic field is zero outside the solenoid and perfectly uniform inside.

Approximating the field as constant in direction and magnitude near the center of the solenoid allows us to use Ampere's Law to calculate its magnitude.

Applying Ampere's Law to a solenoid.

Choosing a flat rectangle as the Amperian loop, we see that the contributions to the loop integral can be broken into four parts. The area bounded by the flat rectangle is penetrated by N loops over a length L.

where n = N/L is the linear density of current loops.

Ampere's Law

Ampere's Law:

The magnetic field in space around an electric current is proportional to the electric current, which serves as its source, just as the electric field in space is proportional to the charge that serves as its source. Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.

In the electric case, the relation of field to source is quantified in Gauss's Law, which is a very powerful tool for calculating electric fields.

Physics Practice MCQs

1-The relation (B. L)I=UI is called
a-Amperes law b-Ohm’s law
c-Gauss’s law d-Newton’s law

2-(B. L)i is equal to
a-UI b=UNI
c-UNI/2r d-U/I

3-The magnetic field of induction of Toroid is
a-UNI/2r b-UI/2r
c-U2r/NI d-2r/UI

4-The magnetic field of induction of solenoid is
a-Uin b-UI
c-Uin/2r d-2r/UnI

5- is a coil of an insulated copper wire
wound on a long cylinder.
a-solenoid b-Toroid
c-Battery cell d-Hollow cylinder

6- is a coil of an insulated copper wire
wound on a circular shape material.
a-Toroid b-Solenoid
c-Circuit d-Potentiometer

7-The dot product between tangential components
And the summation of all the components of length
is equal to product of and
a-U and I b-U and N
c-U and V d-U and R

8- (B. L)=UI here U is called
a-Permeability of free space
b-Permitivity of free space
c-Relative permitivity
d-Di-electric constant

9-The value of U is
a-4*10^-7weber/A.m b-4*106-7weber/A.m
c-4*10^-7weber/C.m d-4*10^7coulomb/weber.m

10-The unit of B is
a-Weber b-Coulomb
c-Ampere d-Ohm

11-Acording to Biot and Savart law the magnetic field
of induction is directly related with
a-Twice of the current.
b-Twice of the voltage.
c-Twice of the resistance.
d-Thrice of the voltage.

12-What does the relation B2I/r represents.
a-Biot and Savart law b-Faraday law
c-Lenz law d-Inverse square law

13-Acording to Biot and Savart law the magnetic field
of induction B is related to
a-1/r b-r
c-r^2 d-i/r^2

14-Acording to Biot and Savart law the equation
B=(constant)2I/r here constant is
a-U/4 b-4U/
c-U/4r d-4r/U

15- l=Circumference of a circle which is equal to
a-2r b-2r^2
c-r^2 d-4/3 r^3

16-The magnetic field of induction will be maximum
If
a-B and L are in same direction.
b-B and L are in opposite direction.
c-B and L are perpendicular.
d-B is zero

17-Ampere’s law was proposed by
a-Ampere b-Faraday
c-Newton d-Coulomb

18-Ampere’s law carries which variable quantity.
a-I b-V
c-R d-C

19-The magnetic field of induction will be minimum if
a-B and L are in same direction.
b-B and L are in opposite direction.
c-B and L are perpendicular.
d-B is zero

20-Magnetic flux are the lines of forces around a
a-charge b-magnet
c-electron d-positron

Static Method

Static Method:

A method may be declared as static, meaning that it acts at the class level rather than at the instance level.