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Unformatted text preview: Solution for Homework 15 Adding Magnetic Fields Solution to Homework Problem 15.1(The Magnetic Field Direction Due to Two Current- Carrying Wires at a Given Point) Problem: The figure shows two current-carrying wires, one current pointing out of the page and one pointing into the page. The wire carry currents of equal magnitude. A point p is located halfway between the two wires. What is the direction of the total magnetic field at point p due to the two current-carrying wires? Select One of the Following: (a-Answer) The magnetic field points upward. (b) The magnetic field points downward. (c) The magnetic field points to the right. (d) The magnetic field points to the left. p Solution (a) The magnetic field due to the lefthand wire at point p can be found using the right-hand rule to have a component upward and a component to the right. (b) The magnetic field due to the righthand wire at point p can be found using the right-hand rule to have a component upward and a component to the left. (c) Because point p is midway between the two current-carrying wires, the magnetic fields are equal in magnitude. Thus, the magnetic field components to the right and left cancel. This leaves a magnetic field with a component only upward. Total Points for Problem: 3 Points Solution to Homework Problem 15.2(Magnetic Field of Segment of Loop) Problem: A segment of a circular loop of wire lies in the y- z plane. It occupies the 1 4 plane with y > and z > , so the loop forms one quarter of a circle. The wire carries a current I . Compute the magnetic field at the origin. Select One of the Following: (a) μ I 2 πR ˆ x (b)- μ I 2 πR ˆ x (c) μ I 8 R ˆ x (d-Answer)- μ I 8 R ˆ x (e) y z I R 1 Solution y z I x out of page B in to page at origin R Definitions vector B ≡ Magnetic Field at Origin I ≡ Current in Loop R ≡ Radius of Loop vector ℓ ≡ Vector in direction of current Strategy: Integrate the Biot-Savart Law over the segment of the loop. (a) Write Magnetic Field as Sum: Using the Law of Linear Superposition for the magnetic field, the field at the origin can be written as the sum of the fields generated by the infinitesimal elements of current around the loop, vector B = summationdisplay d vector B. (b) Use Biot-Savart Law: The field of an infinitesimal current element is given by the Biot-Savart Law, d vector B = μ 4 π I vector d vector ℓ × ˆ R R 2 where vector ℓ points in the direction of the current. (c) Use Right Hand Rule: Using the right hand rule on Id vector ℓ × ˆ R shows the direction of the magnetic field to be into the page which is the- ˆ x direction. (You also have to use the right hand rule to work out whether the x-axis is into or out of the page), so Id vector ℓ × ˆ R =- Idℓ ˆ x for the current as drawn in the figure....
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