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K L University 1

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2 MAGNETOSTATICS

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3 Introduction to Magneto statics – Magnetic field, Magnetic force, Magnetic flux Biot-Savat’s law -- Applications of Bio-Savart’s law Ampere’s Circular Law Cyclotron Hall Effect & Applications LCR Series Resonance Circuit Worked Problem

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4 Magnet and Magnetic Field:

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6 –The magnitude of dB is inversely proportional to r 2, where r is the distance from the element ds to the point P. –The magnitude of dB is proportional to the current I and to the length ds of the element. –The magnitude of dB is proportional to sin ϕ, where ϕ is the angle between the vectors ds and r hat. Biot-Savart law:

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7 µ o is a constant called the permeability of free space; µ o =4· x 10 -7 Wb/A·m (T·m/A) Biot-Savart law gives the magnetic field at a point for only a small element of the conductor ds. To determine the total magnetic field B at some point due to a conductor of specified size, we must add up every contribution from all elements ds that make up the conductor (integrate)!

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8 The direction of the magnetic field due to a current carrying element is perpendicular to both the current element ds and the radius vector r hat. The right hand rule can be used to determine the direction of the magnetic field around the current carrying conductor: –Thumb of the right hand in the direction of the current. –Fingers of the right hand curl around the wire in the direction of the magnetic field at that point.

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10 Lorentz Force: Charges moving in a magnetic field experience an electromagnetic force.

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11 Magnetic Field of a Thin Straight Conductor:

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12 Magnetic Field of a Thin Straight Conductor: The magnetic field lines are concentric circles that surround the wire in a plane perpendicular to the wire. The magnitude of B is constant on any circle of radius a. The magnitude of the magnetic field B is proportional to the current and decreases as the distance from the wire increases.

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13 Magnetic Field on the Axis of a Circular Current Loop

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14 Electric currents create magnetic fields.

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15 What Is The Hall Effect? According to Hall effect When a magnetic field is applied perpendicular to a current carrying conductor, a potential difference is developed between the points on opposite side of the conductor. http://www.nikhef.nl/pub/linde/MEDIA/ANIMATIONS/FLASH/RemcoBrantjes/hall-effect.swf

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17 When a current-carrying conductor is placed in a MF, a voltage is generated in a direction perpendicular to both the current and the MF. The Hall Effect results from the deflection of the charge carriers to one side of the conductor as a result of the magnetic force experienced by the charge carriers. The arrangement for observing the Hall Effect consists of a flat conducting strip carrying a current I in the x-direction.

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18 A uniform magnetic field B is applied in the y-direction. If the charge carriers are electrons moving in the negative x- direction with a velocity v d, they will experience an upward magnetic force F B.

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19 The electrons will be deflected upward, making the upper edge negatively charged and the lower edge positively charged. The accumulation of charge at the edges continues until the electric field and the resulting electric force set up by the charge separation balances the magnetic force on the charge carriers (F mag = F electric ).

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20 When equilibrium is reached, the electrons are no longer deflected upward. A voltmeter connected across the conductor can be used to measure the potential difference across the conductor, known as the Hall voltage V H. When the charge carriers are positive, the charges experience an upward magnetic force q·(v x B). The upper edge of the conductor becomes positively charged, leaving the bottom of the conductor negatively charged. The sign of the Hall voltage generated is opposite the sign of the Hall voltage resulting from the deflection of electrons.

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21 The sign of the charge carriers can be determined from the polarity of the Hall voltage. When equilibrium is reached between the electric force q·E and the magnetic force q·v d ·B, the electric field produced between the positive and negative charges is referred to as the Hall field, E H, therefore, q·E H = q·v d ·B. E H = v d ·B If d is taken to be the width of the conductor, then the Hall voltage V H measured by the voltmeter is:

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22 The measured Hall voltage gives a value for the drift velocity of the charge carriers if d and B are known. The number of charge carriers per unit volume (charge density), n, can also be determined by measuring the current in the conductor: Area A = thickness t·d, therefore :

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23 Hall coefficient, R H = The Hall coefficient can be determined from The sign and magnitude of R H gives the sign of the charge carriers and their density. In most metals, the charge carriers are electrons.

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25 Principle of operation Particle acceleration is achieved using an RF field between “dees” with a constant magnetic field to guide the particles

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27 Construction:

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28 Working : +ve ions emiited (source) --- accelerated in the gap towards the dee (which is –ve at that time) say D2 Since there is no electric field inside the dees, the +ve ions move with constant velocity along the circles of constant radius ( under the influence of magnetic field) If by the time the ions emerge from D2, the polarity of the applied potential is reversed (D1 is negative). +ve ion again accelerated by the field in the gap. Since their velocity is increased they will move through D1 along circular arc of greater radius.

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29 Cyclotron The +ve ions move faster and faster moving in ever-expanding circles until they reach the outer edge of the dees where they are deflected by deflector plate and strike the target. The time required for the positive ions to make one complete turn within dees is the same for all speeds and is equal to the time period of oscillator. Working :

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30 Alternating E-field Top View Side View Ejected ions Uniform B-field region Injected ions www.hyperphysics.com DEMO Video

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31 Limitation s: The maximum available particle energy is limited due to the following factors: 1)Due to the limited power and frequency of the oscillator. 2)Due to the maximum strength of the magnetic field which can be produced 3)The energy of charged particle emerging from cyclotron is limited due to variation of mass with velocity.

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