Tiêu đề 05. MAGNETISM AND MATTER.pdf ✅

The poles of the same magnet do not come to meet each other due to attraction. They are maintained we cannot get two isolated poles by cutting the magnet from the middle. The other end becomes pole of opposite nature. So, ‘N’ and ‘S’ always exist together. Bar Magnet as an Equivalent Solenoid By calculating the axial field of a finite solenoid carrying current, a bar magnet can be demonstrated as a solenoid. Consider a solenoid of radius a and length 2l with n number of turns per unit length that has current I passing through the solenoid. Considering a small element of thickness dx of the solenoid at a distance x from O such that OP = r. Bar magnet as an Equivalent Solenoid Magnetic field due to n turns at the axis of the solenoid dB = μ0ndxIa 2 2[(r−x) 2+a2] 3/2 Integrating x from -I to +I to get the magnitude of the total field B=μ0nIa 2 2 ∫ dx [(r−x) 2+a2] 3 2 I −I [(r − x) 2 +a 2 ] 3/2 = r 3 And B=μ0nIa 2 2 ∫ dx I −I Therefore B = μ0 4π 2m r 3 From the above expression, it is understood that the magnetic moment of a bar magnet is equal to the magnetic moment of a solenoid. Difference between a Bar Magnet and a Solenoid The bar magnet is a permanent magnet, whereas a solenoid is an electromagnet i.e., it acts as a magnet only when an electric current is passed through. When a bar magnet is cut into two halves, both the pieces act as a magnet with the same magnetic properties, whereas when a solenoid is cut into two halves, it will have weaker fields. The poles of the bar magnet are fixed, whereas, for a solenoid, the poles can be altered. The strength of the magnetic field of a bar magnet is fixed, i.e., unaltered, whereas the strength of the magnetic field of a solenoid depends on the electric current that is passed through it. The Dipole in a Uniform Magnetic Field if we place iron filings around a bar magnet on a sheet of paper and tape the sheet, the fillings rearrange themselves to form a specific pattern. The pattern of iron filings here denotes the magnetic field lines generated due to the magnet. These magnetic field lines give us an approximate idea of the magnetic field B. But many times, we are required to determine the magnitude of the magnetic field B accurately. We accomplish this by placing a small compass needle of known magnetic moment m and moment of inertia and allowing it to oscillate in the magnetic field. The torque on the needle can be given as, τ⃗ = ⃗m⃗⃗⃗⃗ × B⃗⃗ The magnitude of this torque is given by mB sinθ. Here τ is the restoring torque, and θ is the angle between the direction of the magnetic moment (m) and the direction of the magnetic field (B). At equilibrium, we can say that, Id 2θ dt ≅ −mBsinθ The negative sign in the above expression mB sinθ brings us to the conclusion that the restoring torque acting here acts in the opposite direction to the deflecting torque. Also, as the value of θ is very small in radians, we can approximate sin θ ≈ θ. Thus, using this approximation, we can write Id 2θ dt ≅ −mBθ d 2θ dt ≅ − mBθ I The above equation is a representation of a simple harmonic motion and the angular frequency can be given as, ω = mB I and thus, the time period can be stated as, T = 2π√ I mB An expression for magnetic potential energy can also be obtained on lines similar to electrostatic potential energy. The magnetic potential energy Um is given by Um = ∫ τ (θ)d θ = − ∫mB sinθ dθ cosθ = −m B cos θ =-m.B

(a) The magnetic meridian at a place is not a line but a vertical plane passing through the axis of a freely suspended magnet, i.e., it is a plane which contains the place and the magnetic axis. (b) The geographical meridian at a place is a vertical plane which passes through the line joining the geographical north and south, i.e., it is a plane which contains the place and earth’s axis of rotation, i.e., geographical axis. (c) The magnetic Equator is a great circle (a circle with the center at earth’s Centre) on earth’s surface which is perpendicular to the magnetic axis. The magnetic equator passing through Trivandrum in South India divides the earth into two hemispheres. The hemisphere containing south polarity of earth’s magnetism is called the northern hemisphere (NHS) while the other, the southern hemisphere (SHS). (d) The magnetic field of earth is not constant and changes irregularly from place to place on the surface of the earth and even at a given place it varies with time too. Elements of earths magnetism The magnetism of earth is completely specified by the following three parameters called elements of earth’s magnetism (a) Variation or Declination At a given place the angle between the geographical meridian and the magnetic meridian is called declination, i.e., at a given place it is the angle between the geographical north-south direction and the direction indicated by a magnetic compass needle, Declination at a place is expressed at q° E or q°W depending upon whether the north pole of the compass needle lies to the east (right) or to the west (left) of the geographical north-south direction. The declination at London is 10oW means that at London the north pole of a compass needle points 10°W, i.e., left of the geographical north. GM  MM (A) Declination =  oW oE W E S N   (B) (b) Inclination or Angle of Dip f: It is the angle which the direction of resultant intensity of earth’s magnetic field subtends with horizontal line in magnetic meridian at the given place. Actually, it is the angle which the axis of a freely suspended magnet (up or down) subtends with the horizontal in magnetic meridian at a given place. Here, it is worthy to note that as the northern hemisphere contains south polarity of earth’s magnetism, in it the north pole of a freely suspended magnet (or pivoted compass needle) will dip downwards, i.e., towards the earth while the opposite will take place in the southern hemisphere. BV BH BI  Magnetic meridian (A) Dip =  Magnetic meridian Axis of a freely suspended magnet (B) Horizontal line  Angle of dip at a place is measured by the instrument called Dip-Circle in which a magnetic needle is free to rotate in a vertical plane which can be set in any vertical direction. Angle of dip at Delhi is 42°. (c) Horizontal Component of Earth’s Magnetic Field BH : At a given place it is defined as the component of earth’s magnetic field along the horizontal in the magnetic meridian. It is represented by BH and is measured with the help of a vibration or deflection magnetometer. At Delhi the horizontal component of the earth’s magnetic field is 35 MT, i.e., 0.35 G. If at a place magnetic field of earth is BI and angle of dip f, then in accordance with figure (a). BH = BI cos f and Bv = BI sin f .... (1) so that, tan f = Bv BH and Ι = √BH 2 + Bv 2 ....(2) Magnetization and Magnetic Intensity The earth abounds with a bewildering variety of elements and compounds. In addition, we have been synthesizing new alloys, compounds and even elements. One would like to classify the magnetic properties of these substances. In the present section, we define and explain certain terms which will help us to carry out this exercise. We have seen that a circulating electron in an atom has a magnetic moment. In a bulk material, these moments add up vectorially and they can give a net magnetic moment which is non-zero. Magnetization Magnetization M of a sample to be equal to its net magnetic moment per unit volume: M = mnet V M is a vector with dimensions L–1 A and is measured in units of A m–1 . Consider a long solenoid of n turns per unit length and carrying a current I. The magnetic field in the interior of the solenoid was shown to be given by B0 = μ0 nI If the interior of the solenoid is filled with a material with non- zero magnetization, the field inside the solenoid will be greater than B0. The net B field in the interior of the solenoid may be expressed as B = B0 + Bm where Bm is the field contributed by the material core. It turns out that this additional field Bm is proportional to the magnetization M of the material and is expressed as Bm = μ0M where μ0 is the same constant (permeability of vacuum) that appears in Biot-Savart’s law.