PES 1120 Spring 2014, Spendier Lecture 27/Page 1

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1 PES 1120 Spring 2014, Spendier Lecture 27/Page 1 Today: - Magnetic Force on a current-carrying Wire - Torque on a current loop - Magnetic dipole moment - Next Week Monday Review, Wednesday EXAM 2 Last time we studied the consequences and applications of a moving charged particle in magnetic (and electric) fields. Magnetic Force on a Current-Carrying Wire We have seen that a charged particle moving through a magnetic field experiences a magnetic force F B. Since electric current consists of a collection of charged particles in motion, when placed in a magnetic field, a current-carrying wire will also experience a magnetic force. Consider a long straight wire suspended in the region between the two magnetic poles. The magnetic field points out the page and is represented with dots ( ). It can be readily demonstrated that when a downward current passes through, the wire is deflected to the left. However, when the current is upward, the deflection is rightward, as shown in Figure To calculate the force exerted on the wire, consider a segment of wire of length l F qvb sin f l F q B sin f t q is the net charge through length l q F lbsinf t F IlB sin f F I l B

2 PES 1120 Spring 2014, Spendier Lecture 27/Page 2 RHR for Force on Wire: Start fingers in direction of I so that they can curl towards B, thumb points in direction of F. Magnetic Force on a Current : One common use of this force is in speakers. General Calculations For a wire of arbitrary shape, the magnetic force can be obtained by summing over the forces acting on the small segments that make up the wire. A curved wire has infinitely many different tangent lines length vectors dl associated with each point. The force at each point on the wire: df I dl B Net force on wire is found by adding (vectorially) the force at each point together. F I dl B Force on a current loop: Current flowing around a closed path is called a current loop. Shown is a square current loop in an upwards magnetic field. What is the magnitude and direction of the force acting on side 1, 3, 2, and 4? What is the total force acing on the loop?

3 PES 1120 Spring 2014, Spendier Lecture 27/Page 3 RHR for Force on Wire: Start fingers in direction of I so that they can curl towards B, thumb points in direction of F. The net force on this current loop is zero. The net force on any shape current loop in a uniform magnetic field is zero! Torque on a Current Loop But what about the torque on the current loop? The net torque is non-zero. This loop would rotate In an external, uniform magnetic field, a current loop experiences zero net force but a non-zero torque. Even though the net force on the loop vanishes, the forces F 1 and F 3 will produce a torque which causes the loop to rotate about the x-axis. From Physics 1: t r F (r lever arm and at end of lever arm force is applied)

4 PES 1120 Spring 2014, Spendier Lecture 27/Page 4 The torque with respect to the center of the loop is t r F r F r F r F r F r F Application DC Motor We can now understand how a DC motor works. We put a current loop between the poles of a magnet. The torque on the loop causes rotation. We have to switch the direction of the current every time it rotates to keep it rotating in one direction.

5 PES 1120 Spring 2014, Spendier Lecture 27/Page 5 Area vector: It is convenient to introduce the area vector A Anˆ where ˆn is a unit vector in the direction normal to the plane of the loop. The direction of the positive sense of ˆn is set by the conventional right-hand rule. The direction of ˆn is given by curling the fingers in the direction of the current and the thumb pointing in the direction of ˆn. Magnetic Dipole Moment: Using the area vector, there is one more physical quantity which is worth mentioning, it is the magnetic dipole moment defined as μ. m I A [A m 2 ] Thee magnetic dipole moment is a vector, m! Again, the right-hand rule determines the direction of the magnetic moment of a current-carrying loop. This is also the direction of the loop's area vector. We can now rewrite the torque on a current loop as: t I AB mb t ( IA) B sin q mb sin q θ = angle between m (or A ) and B

6 PES 1120 Spring 2014, Spendier Lecture 27/Page 6 Magnetic Force on a Magnetic Moment Now, the atoms of iron (and other metals) have a magnetic moment because of the current loops of the electrons. In non-magnetized objects, they are randomly oriented. But in a magnet, they are (mostly) aligned with each other, giving a net total magnetic moment. In a magnetic field the torque on a bar magnet tends to align the magnet's dipole moment with the direction of the B field. Magnetic Dipoles in a magnetic field rotate to make θ = 0. For a non-magnetized object, like a nail, the field polarizes (rotates) the moments in the direction of the field. Then the nail has a net magnetic moment and is attracted to the magnet. When the field is removed, the atoms randomize their magnetic moments again, and the object returns to be non-magnetized. UNLESS, you heat and then cool it, or shock it by hitting it with a hammer

7 PES 1120 Spring 2014, Spendier Lecture 27/Page 7 Potential Energy: The equation for torque exerted on an magnetic dipole moment m in the presence of an magnetic field B t m B is analogous to t pe, the torque exerted on an electric dipole moment p in the presence of an electric field E. Recalling that the potential energy for an electric dipole is U p E, a similar form is expected for the magnetic case. U m B Example: What is the change in energy when a dipole with μ = 1 A m 2 flips from being opposite to a B = 1 T external field to being parallel to it? Hint: μb = (1A m 2 )(1 T) = 1J.

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