Physics MAGNETISM AND GAUSS’S LAW,THE EARTH’S MAGNETISM, MAGNETIC DECLINATION AND DIP FOR CBSE-NCERT 2
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MAGNETISM AND GAUSS’S LAW,THE EARTH’S MAGNETISM, MAGNETIC DECLINATION AND DIP FOR CBSE-NCERT 2

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`color{blue}{star}` MAGNETISM AND GAUSS’S LAW
`color{blue}{star}` THE EARTH’S MAGNETISM
`color{blue}{star}` MAGNETIC DECLINATION AND DIP

MAGNETISM AND GAUSS’S LAW

`color{blue} ✍️` We know the fact that no net charge is enclosed by the surface. However, in the same figure, for the closed surface `color{blue}((i))` , there is a net outward flux, since it does include a net (positive) charge.

`color {blue}{➢➢}` The situation is radically different for magnetic fields which are continuous and form closed loops. Examine the Gaussian surfaces represented by `color{blue}((i))` or `color{blue}((ii))` in Fig 5.3(a) or Fig. 5.3(b). Both cases visually demonstrate that the number of magnetic field lines leaving the surface is balanced by the number of lines entering it. The net magnetic flux is zero for both the surfaces. This is true for any closed surface.



`color{blue} ✍️` Consider a small vector area element `color{blue}(ΔS)` of a closed surface S as in Fig. 5.6. The magnetic flux through `color{blue}(vec A S)` is defined as `color{blue}(Δphi_B = B * ΔS),` where `color{blue}(B)` is the field at `color{blue}(ΔS).` We divide `color{blue}(S)` into many small area elements and calculate the individual flux through each. Then, the net flux `color{blue}(phi B)` is,

`color{blue}(phi_B = sum_('all') Delta phi_B= sum_('all') B Delta S =0)`

.................(5.9)

`color {blue}{➢➢}` where ‘all’ stands for ‘all area elements `color{blue}(ΔS′)`. Compare this with the Gauss’s law of electrostatics. The flux through a closed surface in that case is given by

`color{blue}(sum E Delta S= q/(ε_0))`

`color {blue}{➢➢}` where `color{blue}(q)` is the electric charge enclosed by the surface. The difference between the Gauss’s law of magnetism and that for electrostatics is a reflection of the fact that isolated magnetic poles (also called monopoles) are not known to exist.

`color{blue} ✍️` There are no sources or sinks of `color{blue}(B)`; the simplest magnetic element is a dipole or a current loop. All magnetic phenomena can be explained in terms of an arrangement of dipoles and/or current loops.

`color {blue}{➢➢}`Thus, Gauss’s law for magnetism is:

`color {blue}{➢➢}` The net magnetic flux through any closed surface is zero.
Q 2617112080

Many of the diagrams given in Fig. show magnetic field lines (thick lines in the figure) wrongly. Point out what is wrong with them. Some of them may describe electrostatic field lines correctly. Point out which ones.

Solution:

(a) Wrong. Magnetic field lines can never emanate from a point, as shown in figure. Over any closed surface, the net flux of B must always be zero, i.e., pictorially as many field lines should seem to enter the surface as the number of lines leaving it. The field lines shown, in fact, represent electric field of a long positively charged wire. The correct magnetic field lines are circling the straight conductor,

(b) Wrong. Magnetic field lines (like electric field lines) can never cross each other, because otherwise the direction of field at the point of intersection is ambiguous. There is further error in the figure. Magnetostatic field lines can never form closed loops around empty space. A closed loop of static magnetic field line must enclose a region across which a current is passing. By contrast, electrostatic field lines can never form closed loops, neither in empty space, nor when the loop encloses charges.

(c) Right. Magnetic lines are completely confined within a toroid. Nothing wrong here in field lines forming closed loops, since each loop encloses a region across which a current passes. Note, for clarity of figure, only a few field lines within the toroid have been shown. Actually, the entire region enclosed by the windings contains magnetic field.

(d) Wrong. Field lines due to a solenoid at its ends and outside cannot be so completely straight and confined; such a thing violates Ampere’s law. The lines should curve out at both ends, and meet eventually to form closed loops

(e) Right. These are field lines outside and inside a bar magnet. Note carefully the direction of field lines inside. Not all field lines emanate out of a north pole (or converge into a south pole). Around both the N-pole, and the S-pole, the net flux of the field is zero.

(f) Wrong. These field lines cannot possibly represent a magnetic field. Look at the upper region. All the field lines seem to emanate out of the shaded plate. The net flux through a surface surrounding the shaded plate is not zero. This is impossible for a magnetic field. The given field lines, in fact, show the electrostatic field lines around a positively charged upper plate and a negatively charged lower plate. The difference between Fig. [(e) and (f)] should be carefully grasped.

(g) Wrong. Magnetic field lines between two pole pieces cannot be precisely straight at the ends. Some fringing of lines is inevitable. Otherwise, Ampere’s law is violated. This is also true for electric field lines.
Q 3134191052

(a) Magnetic field lines show the direction (at every point) along which
a small magnetised needle aligns (at the point). Do the magnetic field lines also represent the lines of force on a moving charged particle at every point?
(b) Magnetic field lines can be entirely confined within the core of a toroid, but not within a straight solenoid. Why?
(c) If magnetic monopoles existed, how would the Gauss’s law of magnetism be modified?
(d) Does a bar magnet exert a torque on itself due to its own field?
Does one element of a current-carrying wire exert a force on another element of the same wire?
(e) Magnetic field arises due to charges in motion. Can a system have
magnetic moments even though its net charge is zero?
Class 12 Chapter example 7
Solution:

(a) No. The magnetic force is always normal to B (remember magnetic force `= qv × B)`. It is misleading to call magnetic field lines as lines of force.
b) If field lines were entirely confined between two ends of a straight solenoid, the flux through the cross-section at each end would be non-zero. But the flux of field B through any closed surface must always be zero. For a toroid, this difficulty is absent because it has no ‘ends’.

(c) Gauss’s law of magnetism states that the flux of B through any closed surface is always zero `ointB•ds = 0`
If monopoles existed, the right hand side would be equal to the monopole (magnetic charge) `q_m` enclosed by S. [Analogous to Gauss’s law of electrostatics `int_s B•ds = mu_0 q_m` is the (monopole) magnetic charge enclosed by S.]

(d) No. There is no force or torque on an element due to the field produced by that element itself. But there is a force (or torque) on an element of the same wire. (For the special case of a straight wire, this force is zero.)
(e) Yes. The average of the charge in the system may be zero. Yet, the mean of the magnetic moments due to various current loops may not be zero. We will come across such examples in connection with paramagnetic material where atoms have net dipole moment through their net charge is zero.

THE EARTH’S MAGNETISM

`color{blue} ✍️` Earlier we have referred to the magnetic field of the earth. The strength of the earth’s magnetic field varies from place to place on the earth’s surface; its value being of the order of `color{blue}(10^(-5) T)`.

`color {blue}{➢➢}`Originally the magnetic field was thought of as arising from a giant bar magnet placed approximately along the axis of rotation of the earth and deep in the interior. However, this simplistic picture is certainly not correct.

`color {blue}{➢➢}` The magnetic field is now thought to arise due to electrical currents produced by convective motion of metallic fluids (consisting mostly of molten iron and nickel) in the outer core of the earth. This is known as the dynamo effect.

`color {blue}{➢➢}` The magnetic field lines of the earth resemble that of a (hypothetical) magnetic dipole located at the centre of the earth.

`color {blue}{➢➢}` The axis of the dipole does not coincide with the axis of rotation of the earth but is presently titled by approximately `color{blue}(11.3^0)` with respect to the later. In this way of looking at it, the magnetic poles are located where the magnetic field lines due to the dipole enter or leave the earth.

`color {blue}{➢➢}` The location of the north magnetic pole is at a latitude of `color{blue}(79.74^(0) N)` and a longitude of `color{blue}(71.8^0 W)`, a place somewhere in north Canada. The magnetic south pole is at `color{blue}(79.74^0 S, 108.22^0 E)` in the Antarctica.

`color {blue}{➢➢}` The pole near the geographic north pole of the earth is called the north magnetic pole. Likewise, the pole near the geographic south pole is called the south magnetic pole. There is some confusion in the nomenclature of the poles.

`color {blue}{➢➢}` If one looks at the magnetic field lines of the earth (Fig. 5.8), one sees that unlike in the case of a bar magnet, the field lines go into the earth at the north magnetic pole `color{blue}(N_m)` and come out from the south magnetic pole `color{blue}(S_m)`.

`color {blue}{➢➢}` The convention arose because the magnetic north was the direction to which the north pole of a magnetic needle pointed; the north pole of a magnet was so named as it was the north seeking pole. Thus, in reality, the north magnetic pole behaves like the south pole of a bar magnet inside the earth and vice versa.

Q 3164191055

The earth’s magnetic field at the equator is approximately `0.4 G.` Estimate the earth’s dipole moment.
Class 12 Chapter example 8
Solution:

From Eq. (5.7), the equatorial magnetic field is,

`B_E (mu_0m)/(4pir^3)`

We are given that `BE ~ 0.4 G = 4 × 10–5 T. `For r, we take the radius of the earth `6.4 × 106` m. Hence,

`m = (4xx10^(-5) xx(6.4xx10^6)^3)/(mu_0//4pi) = 4 × 10^2 × (6.4 × 10^6)^3`

`(μ0//4π = 10^(–7))`

`= 1.05 × 10^(23) A m^2`
This is close to the value `8 × 10^(22) A m^2` quoted in geomagnetic texts.

Magnetic declination and dip

`color{blue} ✍️` Consider a point on the earth’s surface. At such a point, the direction of the longitude circle determines the geographic north-south direction, the line of longitude towards the north pole being the direction of true north.

`color{blue} ✍️` The vertical plane containing the longitude circle and the axis of rotation of the earth is called the geographic meridian. In a similar way, one can define magnetic meridian of a place as the vertical plane which passes through the imaginary line joining the magnetic north and the south poles.

`color {blue}{➢➢}` This plane would intersect the surface of the earth in a longitude like circle. A magnetic needle, which is free to swing horizontally, would then lie in the magnetic meridian and the north pole of the needle would point towards the magnetic north pole.

`color {blue}{➢➢}` Since the line joining the magnetic poles is titled with respect to the geographic axis of the earth, the magnetic meridian at a point makes angle with the geographic meridian. This, then, is the angle between the true geographic north and the north shown by a compass needle. This angle is called the magnetic declination or simply declination (Fig. 5.9).



`color{blue} ✍️` The declination is greater at higher latitudes and smaller near the equator. The declination in India is small, it being `color{blue}(0^(0) 41′ E)` at Delhi and `color{blue}(0^(0) 58′ W)` at Mumbai. Thus, at both these places a magnetic needle shows the true north quite accurately.

`color{blue} ✍️` There is one more quantity of interest. If a magnetic needle is perfectly balanced about a horizontal axis so that it can swing in a plane of the magnetic meridian, the needle would make an angle with the horizontal (Fig. 5.10).

`color{blue} ✍️` This is known as the angle of dip (also known as inclination). Thus, dip is the angle that the total magnetic field `color{blue}(B_E)` of the earth makes with the surface of the earth. Figure 5.11 shows the magnetic meridian plane at a point `color{blue}(P)` on the surface of the earth. The plane is a section through the earth. The total magnetic field at `color{blue}(P)` can be resolved into a horizontal component `color{blue}(H_E)` and a vertical component `color{blue}(Z_E)`. The angle that `color{blue}(B_E)` makes with `color{blue}(H_E)` is the angle of dip, `color{blue}(I)`.



`color {blue}{➢➢}` In most of the northern hemisphere, the north pole of the dip needle tilts downwards. Likewise in most of the southern hemisphere, the south pole of the dip needle tilts downwards.

`color{blue} ✍️` To describe the magnetic field of the earth at a point on its surface, we need to specify three quantities, viz., the declination `color{blue}(D)`, the angle of dip or the inclination `color{blue}(I)` and the horizontal component of the earth’s field `color{blue}(H_E)`. These are known as the element of the earth’s magnetic field.

`color{blue} ✍️` Representing the verticle component by `color{blue}(Z_E)` we have

`color{blue}(Z_E B_E sin I)`

......................(5.10(a))

`color{blue}(H_E = B_E cosI)`

................(5.10 (b))

`color {blue}{➢➢}`which gives,

`color{blue}(tan I = (Z_E)/(H_E))`

.....................(5.10(c))
Q 2617212180

In the magnetic meridian of a certain place, the horizontal component of the earth’s magnetic field is `0.26G` and the dip angle is `60^0`. What is the magnetic field of the earth at this location?

Solution:

It is given that `H_E = 0.26 G`. From Fig. , we have

`cos60^0 = H_E/B_E`

`B_E = H_E/(cos60^0)`

` = (0.26)/(1/2) = 0.52 G`
 
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