Contents
Studying Physics Topics can lead to exciting new discoveries and technological advancements.
What are the Magnetic Lines of a Magnetic? What do you Mean by Relative Magnetic Permeability?
Magnetic Lines Of Induction
The magnetic lines of force in a uniform magnetic field are equidistant parallel straight lines [Fig.], If a piece of any magnetic material (i.e., iron, steel, nickel, etc.) is placed in this kind of external ‘magnetic field, a magnetic field is induced in the specimen which is converted into a temporary magnet. The external magnetic field is referred as the inducing magnetic field.
Let an iron bar PQ be placed parallel to a uniform magnetic field [Fig.], At the end of PQ through which the magnetic lines of force enter, a S -pole is developed and at the other end, a N -pole is developed. So, due to induction, the magnetic lines of force are rearranged. This rearrangement inside and outside the magnetic material is discussed below.
The lines inside the magnetic material are generated due to superposition of two kinds of lines of force:
- the lines of force due to the inducing magnetic field and
- the lines of magnetisation (the magnetic material is temporarily converted into a magnet due to induction and the lines of the magnetic field of this temporary magnet are called lines of magnetisation).
The lines of force outside the magnetic material are generated due to superposition of two types of lines of forces:
- the lines of force due to inducing magnetic field and
- the lines of force due to temporary magnetic field of the magnetised specimen.
From Fig. we see that, outside the iron-bar, the lines of force suffer bending and inside the bar are densely crowded. In the region adjacent to the points A and B outside the iron bar the lines of force are less dense. Note that, inside the bar PQ, the lines of force of the inducing field and the lines of mangetisation are unidirectional.
It means that inside the bar PQ, the resultant of the two magnetic fields increases and hence in that region and in the adjacent regions C and D, the lines of force are densely crowded. In the regions adjacent to the points A and B, the inducing magnetic field and the temporary magnetic field are oppositely directed and hence the resultant magnetic field decreases there. So the lines of force become less dense in that region.
Definition: The lines of force generated inside a magnetic material due to superposition of the lines of force of the inducing magnetic field and the lines of magnetisation are called the lines of induction.
Outside a magnetic material lines of magnetic induction follow magnetic lines of force. In some materials, called diamagnetic materials, the scenario would be the reverse of the above picture [see section 2.5],
Some Magnetic Quantities And Their Relations
Magnetic permeability of a material:
Let the current through a long straight solenoid = I; number of turns per unit length of the solenoid = n. If air or vacuum is taken as the core of the solenoid, the magnetic field produced along the axis of the solenoid (see section 1.5.1 of the chapter ‘Electromagnetism’),
Bo = µ0nI …… (1)
where, µ0 = magnetic permeability of vacuum (or air)
= 4π × 10-7 H ᐧ m-1
If a rod of any material is introduced inside the solenoid co-axi-ally, the magnetic field along the axis of the solenoid will change; this magnetic field can be expressed as,
B = µnI …….. (2)
This quantity µ is called the magnetic permeability of the mate-rial used.
Relative magnetic permeability: If air or vacuum is replaced by a material, the fractional change of magnetic field in that material is called the relative magnetic permeability of that material.
So, relative magnetic permeability,
µr = \(\frac{B}{B_0}\) = \(\frac{\mu n I}{\mu_o n I}\) = \(\frac{\mu}{\mu_0}\) …. (3)
Unit: Equations (1) and (2) show that, the units of µ0 and µ are the same, i.e., H ᐧ m-1.
Again, equation (3) shows that, being the ratio of two identical quantities µ and µ0, µr has no unit.
Classification of materials: Depending on the value of relative magnetic permeability µ, different materials can be divided into three groups:
i) µr < 1, i.e., µ < µ0 : diamagnetic material;
ii) µr > 1, i.e., µ > µ0 : paramagnetic material;
iii) µr >> 1, i.e., µ >> µ0 : ferromagnetic material.
For example, in case of aluminium, µr = 1.00002, hence it is a paramagnetic material; in case of copper, µr = 0.9999904 , hence it is a diamagnetic material; in case of iron, µr = 1000 to 5000 (approx.), hence it is a ferromagnetic material. These materials are discussed elaborately later (section 2.5).
Magnetisation: If the magnetising field at any point in vacuum or air be \(\vec{H}\), the magnetic field at that point, \(\vec{B}_0\) = \(\mu_0 \vec{H}\).
If vacuum or air at a point is replaced by any other material, magnetism is induced in that material and hence the magnetic field acting will also change from \(\vec{B}_0\) to \(\vec{B}\) (say).
Naturally, \(\vec{B}\) ≠ µ0\(\vec{H}\). If the material is paramagnetic or ferromag-netic, \(\vec{B}\) > µ0\(\vec{H}\). Here, it is assumed that,
\(\vec{B}\) = µ0\(\vec{H}\) + µ0\(\vec{M}\) = µ0(\(\vec{H}\) + \(\vec{M}\))
The first term on the right hand side (µ0\(\vec{H}\)) comes from the magnetising field \(\vec{H}\) produced because of electric current or some other external cause, and the second term (µ0\(\vec{M}\)) comes due to induced magnetism inside the material. So, the term (µ0\(\vec{M}\)) indicates the additional magnetic field produced due to magnetic induction. The quantity \(\vec{M}\) is called the intensity of magnetisation of the material or simply the magnetisation vec-tor, the magnitude of which at a point is given as the net dipole moment per unit volume around that point. By calculation, we can show that (this calculation is omitted here), magnetic moment per unit volume of a material is \(\vec{M}\).
Definition: If unit volume is considered around any point in a material, magnetic moment of that volume is called the intensity of magnetisation at that point.
Unit: From equation (4) it is clear that, the unit of \(\vec{M}\) is the same as the unit of \(\vec{H}\). This unit is A ᐧ m-1 . \(\vec{B}\), \(\vec{H}\) and \(\vec{M}\) are called the three magnetic vectors.
Magnetic susceptibility of a material: In most of the materials, the intensity of magnetisation at a point is directly proportional to the magnetic intensity at that point, i.e.,
M ∝ H or, M = kH
This constant k is called the magnetic susceptibility of the mate-rial. The property by virtue of which a material can be magne-tised is its magnetic susceptibility.
In vector form, \(\vec{M}\) = k\(\vec{H}\) ….. (5)
Now, if H = 1, k = M; from this we can define k.
Definition: Magnetic moment induced per unit volume of a material due to unit magnetic intensity is called the magnetic susceptibility of that material.
Unit: Since the units of M and H are the same, k has no unit.
Mass (magnetic) susceptibility: The ratio of the magnetic susceptibility of a material and its density is called the mass susceptibility of that material. It is denoted by the symbol \(\chi\)
∴ Mass susceptibility, k
\(\chi\) = \(\frac{k}{\rho}\) [where, ρ = density of the material]
Relation between magnetic susceptibility and relative magnetic permeability:
We know that,
\(\vec{B}\) = µ\(\vec{H}\) [see the chapter ‘Electromagnetism’]
\(\vec{B}\) = µ0(\(\vec{H}\) + \(\vec{M}\)) and \(\vec{M}\) = k\(\vec{H}\)
[from equations (4) and (5) ]
Combining them we can write,
µ\(\vec{H}\) = µ0(\(\vec{H}\) + \(\vec{M}\)) = µ0(\(\vec{H}\) + k\(\vec{H}\)) = µ0(1 + k)\(\vec{H}\)
∴ \(\frac{\mu}{\mu_0}\) = 1 + k or, µr = 1 + k
[µr = \(\frac{\mu}{\mu_0}\) = relative magnetic permeability]
or, k = µr – 1 ….. (6)
For vacuum or air, µr = 1, hence, k = 0
Table-1
The values of µr and k of some materials
Hence, for paramagnetic material, k > 0 (positive); for diamagnetic material, k < 0 (negative); for ferromagnetic material, k \(\gg\) 0 (large positive number).
Significance:
i) In case of paramagnetic and diamagnetic materials inagnetic susceptibilities are very low and are respectively positive and negative numbers. It means that, magnetic induction in these materials are very negligible. Thus, these are called non-magnetic materials.
ii) On the other hand, in case of ferromagnetic materials magnetic susceptibilities k are large positive numbers. Hence, in this kind of materials, strong magnetic induction takes place. Thus, these are known as magnetic materials (e.g. iron, nickel, cobalt). Since the value of k is very large, from the relation, \(\vec{B}[latex] = µ0([latex]\vec{H}[latex] + [latex]\vec{M}[latex]) = µ0([latex]\vec{H}[latex] + k[latex]\vec{H}[latex]), we come to know that the term µ0M is much greater than the term µ0H . So, the magnitude of the magnetic field [latex]\vec{B}\) is mainly determined by induced magnetism.
In CGS or Gaussian system: Equations (1) and (2) respectively, can be replaced by the relations,
H0 = 4πni [magnetic permeability of vacuum or air = 1 ]
and H = 4πµni [magnetic permeability of the material = µ]
So, relative magnetic permeability, µr = \(\frac{H}{H_0}\) = µ; so in this system, magnetic permeability and relative magnetic permeability of a material is the same.
Again, in this system equation (4) is written as,
\(\vec{B}\) = \(\vec{H}\) + 4π\(\vec{M}\)
But we know that, \(\vec{B}\) = µ\(\vec{H}\)
So, µ\(\vec{H}\) = \(\vec{H}\) + 4πk\(\vec{H}\) [∵ \(\vec{M}\) = k\(\vec{H}\)]
or, µ = 1 + 4πk or, k = \(\frac{\mu-1}{4 \pi}\)
Magnetic retentivfty and coercivity: when a magnetic material is placed in a magnetic field, the material acquires magnetism due to induction. This magnetism does not vanish ) completely even after the withdrawal of the magnetic field; some amount of magnetism is left behind in the material.
Magnetic retentivity: The property by virtue of which a magnetic material retains some magnetism in it even after withdrawal of the magnetising field, is called the retentivity of that material.
The magnetism, retained in a magnetic material even after the removal of magnetic field applied on it, is called residual mag-netism.
There are some magnetic materials for which the residual mag-netism is almost zero, i.e., they are almost completely demagnetised.
Magnetic coercivity: The property by virtue of which a magnetic material can retain induced magnetism even if used roughly, i.e., subjected to demagnetising forces, is called the coercivity of that material.
Differences between wrought iron and steel on magnetic properties: If two identical rods—one of soft iron and the other of steel—are placed in the same magnetic field, both of them acquire approximately equal amount of magne-tism. If the rods are then removed from the magnetic field, both soft iron and steel retain almost the entire magnetism. Soft iron can hold slightly more magnetism than steel. But if this soft iron magnet is handled roughly, its magnetism dies out easily com-pared to that of steel. So, it can be said that the magnetic reten- tivity of soft iron is slightly greater, but the coercivity of soft iron is much less than that of steel.
A ferromagnetic material can be converted into a strong magnet easily. In case of a ferromagnetic material like soft iron or steel, if a graph of intensity of magnetisation (M) vs magnetic field intensity (H) is drawn, we will get a closed loop [Fig.].
It is known as the magnetisation cycle. In case of steel, the loop is OABCDA and in case of soft iron, the loop is OA’B’C’D’A’. From the graph, it is evident that the intensity of magnetisation (M) is not zero (OB or OB’) even if the magnetic field intensity (H) is reduced to zero from its maximum value, i.e., M lags behind H. This lagging of the intensity of magnetisation is called hysteresis. However, M falls to zero if H is given a certain value (OC or OC’) in the opposite direction.
According to the diagram, OB is the retentivity of steel, OB’ is the retentivity of soft iron, OC is the coercivity of steel and OC’ is the coercivity of soft iron.
Selection of material to construct a permanent magnet: The magnetic material chosen to construct a permanent magnet should have the following properties.
- The material should have high retentivity, so that it can retain sufficient magnetism in it even after withdrawal of the magnetising field.
- The saturation magnetisation of the material should be high enough which can make a strong polarity.
- The material should also have high coercivity so that it can retain induced magnetisation, even if used roughly.
- The magnetic susceptibility of the material should be of high magnitude.
Though all of the above properties do not match properly, steel, rather than soft iron is used to construct permanent magnets. Besides steel, there are some metallic alloys like alnico (Fe 51%, Cu 3%, A1 8%, Ni 14% , Co 24% ); ticonal (Fe 47% , Cu 3%, A1 8%, Ti 2%, Ni 15%, Co 25%), etc. possess the above mentioned qualities and can be used to construct permanent magnets.
Selection of material to construct an electromagnet: To construct an electromagnet, a material which possesses the following properties should be chosen.
- The material should have low retentivity so that it can lose almost all of its magnetism as soon as the applied magne-tising field is withdrawn.
- The saturation magnetisation of the material should be high enough which can make a strong polarity.
- The material must have low coercivity so that it can be eas-ily demagnetised.
- Hysteresis loss for the material should be low so that during magnetisation and demagnetisation the temperature of the material should remain more or less constant. Soft iron or stalloy (an alloy of 5% silicon and 95% iron) possess these qualities and hence can be used as the core of an electro-magnet.
Selection of material as the core of a transformer or dynamo: To prepare the core of a transformer or a dynamo, a material of high magnetic permeability should be chosen. Soft iron possesses such qualities, and hence it is used as the core. Moreover, metallic alloys, like permalloy (50% iron and 50% nickel) and transformer steel (96% iron and 4% silicon), are used nowadays for the same purpose.
Numerical Examples
Example 1.
The number of turns of a solenoid of length 10 cm is 1000. If the air inside it is replaced by a magnetic material and 1A current is passed through the coil, the magnitude of magnetic field at any point on its axis becomes 20 T. Determine the magnetic intensity at that point and relative magnetic permeability of the magnetic material.
Solution:
Number of turns per unit length of the coil,
n = \(\frac{1000}{10}\) = 100 cm-1 = 10000 m-1
∴ Magnetic intensity on the axis,
H = nI = 10000 × 1 = 10000 A ᐧ m-1
If the interior of the solenoid is a vacuum or contains air then magnetic field on the axis,
B0 = µ0nI = (4π × 10-7) × 10000 = 12.56 × 10-3T
Due to the presence of the magnetic material, B = µnI = 20 T
∴ Relative magnetic permeability,
µr = \(\frac{\mu}{\mu_0}\) = \(\frac{B}{B_0}\) = \(\frac{20}{12.56 \times 10^{-3}}\) = 1592
Example 2.
Relative magnetic permeability of a magnetic medium is 1000. If the magnetic field at any point in the medium be 0.1 Wb ᐧ m-2, what will be the values of magnetic intensity and intensity of magnetisation at that point.
Solution:
Here, B = 0.1 Wb ᐧ m-2 and µr = 1000
So, µ = µ0µr = 4π × 10-7 × 1000 = 4π × 10-4 H ᐧ m-1
Now, magnetic intensity, H = \(\frac{B}{\mu}\) = \(\frac{0.1}{4 \pi \times 10^{-4}}\) = 79.6 A ᐧ m-1
Again, B = µ0(H + M)
Intensity of magnetisation,
M = \(\frac{B}{\mu_0}\) – H = \(\frac{0.1}{4 \pi \times 10^{-7}}\) – 79.6
= 79577.5 – 79.6 = 79497.9 A ᐧ m-1
Example 3.
An iron cored toroid has ring radius 7 cm and number of turns 500. If 2 A current is passed through the wire, what will be the value of magnetic field on the axis of the toroid? Given, relative magnetic permeability of iron = 1500.
Solution:
Length of the circular axis of the ring
= 2πr = 2 × \(\frac{22}{7}\) × 7 = 44 cm = 0.44 m
∴ The number of turns per unit length of the toroid,
n = \(\frac{500}{0.44}\)m-1
∴ Magnetic field on the axis of the toroid,
B = µnI = µ0µrnI
= (4π × 10-7) × 1500 × \(\frac{500}{0.44}\) × 2 = 4.28 Wb ᐧ m-2