T08 - TUTORIAL 8 1. In lectures, the magnetic force on a...

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1 TUTORIAL 8 1. In lectures, the magnetic force on a charged particle was given by the empirical expression: F m = Q v ! B ( ) . (i) Use this expression to derive a general expression for the force on a circuit in a magnetic field. Consider the circuit shown in (a) below. B z l F 0 B a n ! w F 0 F 0 (a) (b) axis of rotation (ii) Apply the formula derived in (i) to show that there is no net force on the circuit. There is, however, a mechanical moment of force, i.e., a torque, about the axis of rotation z , on the circuit in (a) given by T = r ! F , where r is the moment arm (of length w in the figure). (iii) Referring to Figure (b), show that the torque is given by: T F = 0 w sin . (iv) Defining a quantity m a = IS n where S wl = , show that the torque can be expressed as T = m ! B .
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2 2. HELMHOLTZ COILS It is often desirable to make the magnetic field uniform over a region of space. Often, it is not practical to use a solenoid, and a Helmholtz coil is used instead. A Helmholtz coil consists of two circular coils of the same radius, with a common axis, separated by a distance chosen to make the second derivative of B vanish at a point on the axis halfway between the coils. (i) Using the notation in the figure below, write an expression for the magnetic flux density on the axis B z z ( ) . (ii) Determine the relationship between b and a in order that the second derivative vanish. (iii) By Taylor expanding B z z ( ) evaluate the uniformity of the field over the region where z ! a / 2 " a / 10 . P z 2 b a N turns N turns 3. (i) Show that the magnitude of the magnetic flux density at point P due to a straight wire of length l and carrying a current I is given by: B = μ 0 I 4 !" cos # 1 $ cos 2 ( ) .
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T08 - TUTORIAL 8 1. In lectures, the magnetic force on a...

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