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What do you Mean By Magnetic Dipole Moment? What do you Mean by Pole-Strength of a Magnet?
Introduction
To come to the subjects of this chapter, we have to recapitulate some matters of the previous chapters:
Electric dipole and electric dipole moment: From the chapter ‘Electric Field’ we know that, two equal but opposite charges +q and -q kept close each other form an electric dipole [Fig.]. Electric dipole moment \(\vec{p}\) of this dipole is defined as :
\(\vec{p}\) = q\(\vec{r}\) ……. (1)
= magnitude of any charge × position vector of charge +q with respect to charge -q.
So, the magnitude of \(\vec{p}\) is p = |\(\vec{p}\)| = qr and the direction is from -q to +q.
Electric field on the axis of an electric dipole: Let p be a point on the axis of an electric dipole [Fig.], If the distance
x of the point P from the mid-point of the dipole be much greater than the length r of the dipole, the electric field at the point P due to that dipole,
\(\vec{E}\) = \(\frac{1}{4 \pi \epsilon_0} \cdot \frac{2 \vec{p}}{x^3}\) …. (2)
Here, ε0 = permittivity of air or vacuum
= 8.854 × 10-12 C2 ᐧ N-1 ᐧ m-2
Torque on an electric dipole in a uniform electric field: \(\vec{E}\) is a uniform electric field [Fig.]; (-q, +q) is an elec-tric dipole whose dipole moment = \(\vec{p}\). Now, the torque acting on the dipole due to the electric field \(\vec{E}\) is,
\(\vec{\tau}\) = \(\vec{p} \times \vec{E}\) …. (3)
In Fig., the direction of \(\vec{\tau}\) is perpen-dicularly downward with respect to the page of the book, which is denoted by the symbol .
Magnetic field on the axis of a current loop: r = radius of a circular conductor of single turn, i.e., radius of the current loop, I = current in that loop [Fig.].
P is any point on the axis of the current loop which is at a distance x from the centre of the loop. From section 1.4.2 in the chapter ‘Electromagnetism’ we know that, magnetic field produced at the point P due to the current loop is,
B = \(\frac{\mu_0 I}{2} \cdot \frac{r^2}{\left(r^2+x^2\right)^{3 / 2}}\)
[µ0 = magnetic permeability of air or vacuum -4π × 10-7 H ᐧ m-1]
If r\(\ll\)x, then B ≈ \(\frac{\mu_0 I}{2} \cdot \frac{r^2}{x^3}\) = \(\frac{\mu_0 I}{2 \pi} \cdot \frac{\pi r^2}{x^3}\) = \(\frac{\mu_0 I}{2 \pi} \cdot \frac{A}{x^3}\)
where, A = πr2 = area of the current loop.
∴ B = \(\frac{\mu_0}{4 \pi} \cdot \frac{2 I A}{x^3}\)
From corkscrew rule we get, the direction of \(\vec{B}\) is along the axis of the loop; for example in Fig., the direction of \(\vec{B}\) at the point P is outward along the axis. Again, taking that direction as the direction of the area A, it can be expressed as \(\vec{A}\).
Therefore, \(\vec{B}\) = \(\frac{\mu_0}{4 \pi} \cdot \frac{2(I \vec{A})}{x^3}\) …. (4)
Torque on a current loop in a uniform magnetic field:
From section 1.7.2 of the chapter ‘Electromagnetism’ we know that, if a current loop of single turn is kept in a uniform magnetic field B , the torque acting on the loop is,
\(\vec{\tau}\) = \(\overrightarrow{I A} \times \vec{B}\) …. (5)
In Fig., the direction of this torque is perpendicularly down-ward with respect to the page of the book, which is denoted by the symbol .
Magnetic Dipole (Magnetic Dipole Moment or Magnetic Moment)
Comparing equations (2) and (4) and at the same time equations (3) and (5) as described in section 2.1, we get,
- electric field \(\vec{E}\) in electrostatics plays the same role as that of magnetic field \(\vec{B}\) in magnetism.
- the role of the quantity \(\frac{1}{4 \pi \epsilon_0}\) in electrostatics is the same as that of the quantity \(\frac{\mu_0}{4 \pi}\) in magnetism.
- the role of electric dipole moment \(\vec{p}\) in electrostatics is the same as the quantity I\(\vec{A}\) related with a current loop in mag-netism.
Inference: Any current loop behaves as a magnetic dipole. The dipole moment of this dipole is,
\(\vec{p}_m\) = I\(\overrightarrow{\boldsymbol{A}}\)
where, I = current through the loop, \(\overrightarrow{\boldsymbol{A}}\) = area vector of the loop.
The magnitude of \(\vec{A}\) is the same as the magnitude of the area of the loop; the direction of \(\vec{A}\) is in the direction of advancement of the screw-head when a right handed screw is rotated in the direction of current I through the loop.
Naturally, if the current loop contains N turns, its magnetic moment becomes \(\vec{p}_m\) = NI\(\vec{A}\).
Unit of magnetic moment:
Unit of pm = unit of I × unit of A
= ampere ᐧ metre2(A ᐧ m2)
In the CGS or Gaussian system Magnetic dipole moment of a current loop, \(\vec{p}_m\) = i\(\vec{A}\) ; if the current loop contains N turns instead of a single turn, \(\vec{p}_m\) = Ni\(\vec{A}\). Here, the units of A, i and pm are cm2, emu of current and emu ᐧ cm2, respec-tively.
∴ 1 emu ᐧ cm2 = 10A × 10-4m2 = 10-3A ᐧ m2
Significance: Any current loop behaves as a magnetic dipole—it means that, a current loop and a magnet having a north and a south pole are qualitatively equivalent. This similar-ity is discussed with the help of the following two examples.
Similarities between a circular conductor and a magnet: Magnetic lines of force near the centre of a circular conductor are almost parallel to each other and they remain perpendicular to the plane of the circle (see Fig. of the chapter ‘Electromagnetism’). So, almost a uniform magnetic field is generated at that region, which acts normal to the plane of the circular conductor.
From the properties of magnetic lines of force of a permanent magnet, we know that, if the circular conductor is replaced by a small permanent magnet in that region, similar lines of force will be obtained [Fig.]. So, we can conclude that a circular current carrying coil behaves as a permanent magnet. Comparing Fig. of the chapter ‘Electromagnetism’ and Fig., we can also get the rule for the determination of polarity of the circular conductor [Fig.]:
- The face of a circular conductor on which the current appears to flow clockwise, develops a magnetic south pole.
- The face of a circular conductor, on which the current appears to flow anticlockwise, develops a magnetic north pole.
With the help of the following experiment, the magnetic property of a circular conductor can be shown.
de la Rive’s floating battery: In a wide test tube some mercury is taken so that it can float upright in water [Fig.].
Some dilute sulphuric acid (H2SO4) is poured into the test-tube above mercury, and zinc and copper plates are dipped in the acid so that a voltaic cell is formed. This is known as a floating battery. Two conducting wires from the two plates are brought outside the test-tube through a cork fitted at the top. These two wires act as two poles of the battery. Now the two ends of a cir-cular coil are joined with those two poles. Usually the coil con-tains several number of turns instead of a single turn.
When the coil is connected with the battery, the whole system begins to oscillate about a vertical axis. At last when the battery comes to rest, the axis of the circular coil sets itself in the north- south direction. From this directive property, it is understood that the current carrying coil behaves as a magnet. With the help of a bar magnet it can also be observed that likes poles repel and unlike poles attract each other.
Similarities between a current carrying solenoid and a magnet: The arrangement of the lines of force shown in Fig. of the chapter ‘Electromagnetism’ is similar to the arrangement of lines of force of a bar magnet. The face, on which the direction of current flow appears clockwise, a magnetic south pole develops, and on the other face of the solenoid, a magnetic north pole develops. So, we can conclude that a current carrying solenoid behaves as a permanent bar magnet.
Experimental demonstration: Magnetic properties of a solenoid can be shown by replacing the circular coil attached with the de la Rive’s floating battery shown in Fig. with a solenoid. In this case also, the directive and the attractive or repulsive property of the solenoid with another magnet is observed. Actually, from the similarities of a magnet and a current loop it can be concluded that—magnetism is not a separate branch of physics, rather it is a part of electricity and this specific subject is known as electromagnetism.
Pole-strength of a Magnet
In modern theory of magnetism, concept of pole-strength of a magnet is not essential; but for the comparison with electrostat-ics and also to get an idea about the old theory of magnetism, we may discuss here about the pole-strength of a magnet.
Let a bar magnet NS of effective length r be kept at an angle θ with the direction of a uniform magnetic field \(\vec{B}\) [Fig.], The effect on N and S poles are equal and opposite;
hence due to the magnetic field \(\vec{B}\), two equal but opposite forces will act on the two poles. The distance between the line of action of the two forces, NC = rsinθ. So, the torque acting on the bar magnet due to these two forces,
\(\tau\) = magnitude of any one force × perpendicular distance between the two forces
= Frsinθ ………. (1)
Again, in the equation (5) of section 2.1, substituting \(\vec{p}_m\) = I\(\vec{A}\) we get,
\(\vec{\tau}\) = \(\vec{p}_m \times \vec{B}\)
Now, from Fig., we get the value of the torque \(\vec{\tau}\),
\(\tau\) = pmBsinθ ……. (2)
Since a current loop and a magnet are identical, from equations (1) and (2) we get,
Frsinθ = pmBsinθ or, F = \(\frac{p_m}{r} B\) = qmB …… (3)
The term qm is known as the pole-strength of a magnet. The value of the pole-strength of each of N and S poles is qm ; con-ventionally the pole-strength of N pole is taken as +qm and that of S pole is taken as -qm.
Definition: The ratio of the magnetic moment of a magnet to its effective length is called the pole-strength of that magnet.
The strength of two poles of a magnet are equal but opposite; the strength of north pole is taken as positive and that of south pole as negative.
Using vector symbol, equation (3) can be written as
\(\vec{p}_m\) = \(q_m \vec{r}\) and \(\vec{F}\) = \(q_m \vec{B}\)
Clearly, these two equations are identical with the equations \(\vec{p}\) = \(\overrightarrow{q r}\) and \(\vec{F}\) = \(q \vec{E}\) in electrostatics. Due to the similarities between the electric field \(\vec{E}\) and magnetic field \(\vec{B}\), we can say that the role of positive and negative charges (±q) in electrostatics is the same as that of the north and south poles (±qm) in electromagnetism.
Unit of pole-strength:
In SI: Unit of qm = \(\frac{\text { unit of } p_m}{\text { unit of } r}\) = \(\frac{\mathrm{A} \cdot \mathrm{m}^2}{\mathrm{~m}}\) = A ᐧ m
InCGS: Unit of qm is emu of current ᐧ cm ;
1 emu of current ᐧ cm = 10 A × 10-2m = \(\frac{1}{10}\) A ᐧ m
Magnetic moment of a magnet: Magnetic moment of a magnet can also be defined from the concept of pole strength.
Definition: The product of the pole-strength of any pole of a magnet and its effective length is called the magnetic moment of that magnet.
If the distance vector from the south pole to the north pole of a bar magnet is \(2 \vec{l}\) and its pole-strength is qm, then magnetic moment, \(\vec{p}_m\) = \(2 q_m \vec{l}\).
Mutual force between two magnetic poles: We know that, for a charge q, the electric field at any point at a distance r from the charge = force acting on unit positive charge placed at that point,
i.e., E = \(\frac{1}{4 \pi \epsilon_0} \cdot \frac{q}{r^2}\) [ε0 = electrical permittivity of vacuum]
From analogy we can say that—magnetic field at any point at a distance r from a magnetic pole of pole-strength qm = force acting on a unit north pole placed at that point,
i.e., B = \(\frac{\mu_0}{4 \pi} \cdot \frac{q_m}{r^2}\) [µ0 = magnetic permiability of vaccum]
If another magnetic pole of pole-strength \(q_m^{\prime}\) is placed at a distance r from qm, from the equation F = qmB we can say that,
force acting on that pole,
F = \(\frac{\mu_0}{4 \pi} \cdot \frac{q_m \cdot q_m^{\prime}}{r^2}\) …. (4)
According to Newton’s third law of motion, it is the mutual force acting between the poles having pole-strengths qm, and \(q_m^{\prime}\). Equation (4) is called Coulomb’s law in magnetism.
i) If both the poles are north poles or both the poles are south poles, the product qm\(q_m^{\prime}\) becomes positive and hence F is also positive. It means that the direction of F is the same as r, i.e., the force F is repulsive.
ii) If the two poles are unlike, the product qm\(q_m^{\prime}\) becomes negative. It means that F is in the direction opposite to r, i.e., the force F is attractive.
From equation (4), we can say that the force acting between two magnetic poles is,
- directly proportional to the product of the pole-strengths of the two poles, and
- inversely proportional to the square of the distance between the two poles.
In electrostatics, the similar law is the Coulombs law:
F = \(\frac{1}{4 \pi \epsilon_0} \cdot \frac{q q^{\prime}}{r^2}\)
In CGS or Gaussian system:
The corresponding relations of electric field E, magnetic intensity H and magnetic force F in vacuum or in air are respectively,
E = \(\frac{q}{r^2}\), H = \(\frac{q_m}{r^2}\) and F = \(\frac{q_m q_m^{\prime}}{r^2}\)