Introduction to Electromagnetism - 17

Administration MIDTERM II Friday, March 24 Time: 3:50-4:50 pm Room: UA1120 Topics: Capacitors Resistors RC Circuits Kirchkoff’s laws Magnetic Force on a straight wire We are interested in two particular cases The force on a straight wire The torque on a loop Magnetic Force on a straight wire FB = I L x B = ILBsin(?) L is a vector that points in the direction of the current Its magnitude is the length L of the segment I is the current B is the magnetic field ? is the angle between B and L Magnetic Force on a straight wire Problem solving technique The magnetic force is zero if The current in the wire is zero The wire is parallel to the magnetic field The magnetic field is zero Otherwise, determine the angle ? between L and B when the two are drawn starting at the same point Find the magnitude of the force from the previous equation Determine the direction of LxB using the right hand rule. Magnetic Force due to a long straight wire Right Hand Rule to find the Direction of B Point the thumb of the right hand in the direction of the current in the wire Curl the fingers inward the palm; the direction that the fingers curl is the direction of the magnetic field lines around the wire Magnetic Field due to a wire Long Derivation Magnetic Field due to a circular loop Magnetic Field due to a circular loop Right Hand Rule to find the direction of the magnetic field due to a circular loop Curl the finger of your right hand inward toward the palm, following the current arounf the loop. The thumb will point in the direction of the field Instructor Franco Gaspari PHY 1040U (Physics for the biosciences) Introduction to Electromagnetism and Optics Lecture 17 March 16, 2007 n=charge density For a single charge but It applies only for a straight wire in a uniform field. If the wire is arbitrarily shaped, we consider the force on a small (straight) segment ds and integrate along the whole wire. Quick Quiz Consider the current in the length of wire shown in the figure below. Rank the points A, B, and C, in terms of magnitude of the magnetic field due to the current in the length element shown, from greatest to least. (a) A, B, C (b) B, C, A (e) An equal field applies (c) C, B, A (d) C, A, B at all these points. Answer: (b). Point B is closest to the current element. Point C is farther away and the field is further reduced by the sin ? factor in the cross product ds × . The field at A is zero because ? = 0. Answer Torque on a current loop Quick Quiz Rank the magnitudes of the torques acting on the rectangular loops shown in the figure below, from highest to lowest. (All the loops are identical and carry the same current.) (a) a, b, c (b) b, c, a (c) c, b, a (d) a, c, b. (e) All loops experience zero torque. Answer: (c). Because all loops enclose the same area and carry the same current, the magnitude of ? is the same for all. For part (c) in the image, ? points upward and is perpendicular to the magnetic field and ? = ?B, the maximum torque possible. For the loop in (a), ? points along the direction of B and the torque is zero. For (b), the torque is intermediate between zero and the maximum value. Answer Sources of Magnetic Fields Remember the experiment with the 2 wires?A current carrying conductor feels the effect of a magnetic field but also produces a magnetic field! Review: These formula describe how a current carrying wire (or loop) feel the effects of a magnetic field. Biot and Savart studied the opposite phenomenon, that is, what kind of magnetic field is produced by a current carrying wire The Biot-Savart Law Biot and Savart found the following relations between I and B at a point P. (1) (2) (3) (4) Permeability of free space Fig 30-4, p.930 Let’s put everything together. Current carrying wires create a magnetic field and feel the effect of one. What happens if we put two current carrying wires close together? From ex.30.1 gives the direction (attractive) Parallel currents attract. Anti-parallel currents repel. DEFINITION OF AMPERE: I=1A if, when passed through 2 parallel wires of infinite length, separation d = 1m, produces a force of 2 x 107 N/m. The Coulomb is defined from the Ampere. We have defined electrical units using mechanical means. Quick Quiz For I1 = 2 A and I2 = 6 A in the figure below, which is true: (a) F1 = 3F2 (b) F1 = F2/3 (c) F1 = F2 Answer: (c). F1 = F2 as required by Newton’s third law. Another way to arrive at this answer is to realize that Equation 30.11 gives the same result whether the multiplication of currents is (2 A)(6 A) or (6 A)(2 A). Answer At the center of the loop, the Magnetic Field is: 20. A current of 17.0 mA is maintained in a single circular loop of 2.00 m circumference. A magnetic field of 0.800 T is directed parallel to the plane of the loop. (a) Calculate the magnetic moment of the loop. (b) What is the magnitude of the torque exerted by the magnetic field on the loop? 29. The magnetic field of the Earth at a certain location is directed vertically downward and has a magnitude of 50.0 ?T. A proton is moving horizontally toward the west in this field with a speed of 6.20 × 106 m/s. (a) What are the direction and magnitude of the magnetic force the field exerts on this charge? (b) What is the radius of the circular arc followed by this proton? 2. A lightning bolt may carry a current of 1.00 × 104 A for a short period of time. What is the resulting magnetic field 100 m from the bolt? Suppose that the bolt extends far above and below the point of observation. 4. Calculate the magnitude of the magnetic field at a point 100 cm from a long, thin conductor carrying a current of 1.00 A. 6. A conductor consists of a circular loop of radius R and two straight, long sections, as shown in the Figure. The wire lies in the plane of the paper and carries a current I. Find an expression for the vector magnetic field at the center of the loop.