106-20102-MT1

# 106-20102-MT1 - Phys 106 First Midterm Examination Page 1...

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Unformatted text preview: Phys 106 First Midterm Examination Page 1 Saturday, 02-Apr, 2011 PROBLEM (1) (25 points) A positive charge Q is uniformly distributed over a quarter circle (one fourth of a circle) of radius R . The charged quarter circle is located symmetrically relative to the x axis, as shown in the figure. The point P is at the origin, and is the center of the quarter circle. Express your answers in terms of the given quantities and fundamental constants as needed. ( sin 45 ° = cos45 ° = 2 2 ) x y R Q P 45° R d Q ! d ! ! d ! E dE x dE y (a) (10 pts) Find the magnitude of the x component of the electric field at the point P . See the charge element dQ in the figure above. dQ = ! d ! = Q ( 2 " R 4 ) ( Rd # ) = 2 Qd # " dE = k e dQ R 2 = 2 k e Qd ! " R 2 E x = dE x ¡ = dE cos ! ¢ " /4 " /4 ¡ = 2 k e Q " R 2 cos ! ¢ " /4 " /4 ¡ d ! E x = 2 k e Q ¡ R 2 sin ¢ [ ] ! ! /4 ! /4 = 2 k e Q ! R 2 sin( ! 4 ) ! sin( ! ! 4 ) " # \$ % & ' E x = 2 k e Q ! R 2 2 2 ! ( ! 2 2 ) " # \$ % & ' = 2 k e Q ! R 2 2 2 + 2 2 " # \$ % & ' E x = 2 k e Q 2 ! R 2 or E x = Q 2 2 ¡ 2 ¢ R 2 (b) (5 pts) Find the magnitude of the y component of the electric field at the point P . E y = dE y ¡ = dE sin ! ¢ " /4 " /4 ¡ = 2 k e Q " R 2 sin ! ¢ " /4 " /4 ¡ d ! E y = 2 k e Q ¡ R 2 ! cos ¢ [ ] ! ! /4 ! /4 E y = ! 2 k e Q ! R 2 cos( ! 4 ) ! cos( ! ! 4 ) " # \$ % & ' = E y = (c) (5 pts) Using the unit vectors, write down an expression for the net electric field ! E at the point P . ! E = E x ˆ i + E y ˆ j where E y = . ! E = 2 k e Q 2 ! R 2 ˆ i or ! E = Q 2 2 ¡ 2 ¢ R 2 ˆ i (d) (5 pts) If a point charge q is located at x = R 2 , the net electric field at the point P becomes zero. Find the charge q in terms of Q and state whether q is positive or negative. The net electric field at P is ! E = ! E 1 + ! E ¡ , where: ! E 1 = field due to point charge q ! E 2 = field due to charged quarter circle ! E = ! E 1 + ! E ¡ = ! E 1 = ! E ¡ k e q ( R ¡ ) ¡ = ¡ k e Q ¡ ! R ¡ q = Q 2 2 ! q is positive Phys 106 First Midterm Examination Page 2 Saturday, 02-Apr, 2011 PROBLEM (2) (25 points) The spherical region a < r < b carries a positive charge per unit volume ! = A r , where A is a positive constant and r is the distance from the center. A positive point charge Q is located at the center ( r = 0) of the cavity. Express your answers in terms of the given quantities and fundamental constants as needed....
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## This note was uploaded on 07/28/2011 for the course PHYS 105 at Middle East Technical University.

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106-20102-MT1 - Phys 106 First Midterm Examination Page 1...

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