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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics

8.02 Spring 2013

Exam One Solutions Name ______________________________________________ Table ______________ Check Section ____L01 Conrad MW 10-12 am ____L02 Paus MW 12 -2 pm ____L03 Paus MW 2 -4 pm ____L04 Tegmark TR 9 -11 am ____L05 Belcher TR 11-1 pm

____L06 Dourmashkin TR 1-3 pm ____L07 Dourmashkin TR 3-5 pm ____L08 Figueroa MW 12 -2 pm ____L09 Gore MW 2-4 pm

Score

Grader

Problem 1 (25 points)

Problem 2 (25 points)

Problem 3 (25 points)

Problem 4 (25 points)

TOTAL

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Problem 1 (25 points): answers without work shown will not be given any credit. Consider a solid non-conducting sphere of radius a carrying a non-uniform positive charge density given by !(r) = !0(r

2 / a2 ) , where !0 is a positive constant with units [C !m"3] . A very

thin non-conducting concentric spherical shell of radius b , with b > a carries a positive charge Q uniformly distributed on the surface.

Determine the direction and magnitude of the electric field in each of the regions: a) r < a , b) a < r < b , and c) r > b . For each region clearly shown your choice of Gaussian surface.

a) r < a :

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b) a < r < b :

c) r > b :

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Problem 2 (25 points): answers without work shown will not be given any credit. Two parallel rings, each of radius R , are separated by a distance R . A positive charge +Q is uniformly distributed around the upper ring and a negative charge !Q is uniformly distributed around the lower ring. Let z be the vertical coordinate, with z = 0 taken to be the center of the lower negatively charged ring. a) What is the direction and magnitude of the electric field at the point A on the vertical axis, a distance z = R / 2 from the center of the lower ring?

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b) What is the potential difference, V (P)!V (") , between a point P located on the vertical axis a distance +z from the center of the lower ring, and infinity? Set V (!) = 0 .

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c) A positively charged point-like object of mass m and charge q is released from rest at a point B at the center of the upper ring, at z = R . What is the speed of the object when it reaches the point A on the vertical axis at the mid-point between the two rings, at z = R / 2 ?

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Problem 3 (25 points): answers without work shown will not be given any credit. Consider an infinite uniformly positively charged horizontal slab with finite thickness t and volume charge density ! . This slab is tangent to a uniformly positively charged sphere with radius R and volume charge density !0 . Let A be the point of tangency, and let B be the mid-point of the slab, let D be the point opposite to A on top-side of the sheet, and let C be the point midway between B and D at a distance 3t / 4 from A . The electric field at A is zero. Let k be an upward pointing unit vector.

a) Determine the direction and magnitude of the electric field at the point B . (Hint: Use the superposition principle.) Clearly shown your work and any choices of Gaussian surfaces.

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b) Determine the direction and magnitude of the electric field at the point C . (Hint: Use the superposition principle.) Clearly shown your work and any choices of Gaussian surfaces.

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c) What is the charge density !0 of the sphere? Express your answer in terms of t , R , ! , and !0 as needed.

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d) A point-like positively charged object of mass m and charge q is placed at point D and released from rest. Determine the direction and magnitude of the electric force on the point-like object when it is a very far distance d away from the slab, where d >> t and d >> R . Express your answer in terms of !0 , ! , m , q , t , R , d , !0 , and k as needed.

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Problem 4 of 4: (25 points) Concept Questions (Parts A through E) Part A (5 points):

A portion of a circular arc carries a uniform charge per unit length . The arc subtends an angle (in radians) and has radius a, as shown. The total charge on the arc is Q a = . The vertical component of the electric field at point P, the center of the arc, is given by which expression below?

1. cos2 2oa

2. 2 cos2 2oa

3. sin2 2oa

4. 2 sin2 2oa

Answer: 3

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Part B (5 points):

An observer sits at point P a distance a away from a charge Q and a distance 2a away from a change -2Q. The two charges and the observer all lie on the same line, as shown. If we take the electric potential to be zero at infinity, then which of the following statements is true about the electric potential and electric field at point P? Assume Q > 0.

1. The electric potential at P is less than zero and the electric field at P is equal to zero.

2. The electric potential at P is less than zero and the electric field at P points to the right.

3. The electric potential at P is less than zero and the electric field at P points to the left.

4. The electric potential at P is equal to zero and the electric field at P is equal to zero.

5. The electric potential at P is equal to zero and the electric field at P points to the right.

6. The electric potential at P is equal to zero and the electric field at P points to the left.

7. The electric potential at P is greater than zero and the electric field at P is equal to zero.

8. The electric potential at P is greater than zero and the electric field at P points to the right.

9. The electric potential at P is greater than zero and the electric field at P points to the left.

Answer: 5

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Part C (5 points):

Six charges all have a charge of either +1 C or -1 C. They are arranged as shown in the grass seeds representation of the field above. The top charge has a charge of +1 C. What is the total charge of the six charges?

1. +4 C

2. +3 C

3. +2 C

4. +1 C

5. +0 C

6. -1 C

7. -2 C

8. -3 C

9. -4 C Answer: 5

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Part D (5 points): Three similar charged particles are placed one meter apart with the number of units of charge and the sign (+, -) of the charge indicated. Each charge is subject to electric forces caused by other charged particles.

Which of the arrows below best represents the direction of the net force on charge C?

(a) ( b) ( c) (d) (e) none of these Answer: a

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Part E (5 points): The area vectors d

!A at each point on a closed surface (i.e., a surface that surrounds a

volume) are always chosen to point out of the enclosed volume. A closed imaginary surface is called a Gaussian surface. The imaginary Gaussian surface below is a cylinder. A positive charge is located on the cylinder axis above the Gaussian cylinder, as shown below.

Which statement is correct about the flux through surface B and through the entire closed surface A+B+C?

1. The flux through surface B is positive and through the entire closed surface it is positive.

2. The flux through B is positive and through the entire surface it is negative.

3. The flux through B is positive and through the entire surface it is zero.

4. The flux through B is negative and through the entire surface it is positive.

5. The flux through B is negative and through the entire surface it is negative.

6. The flux through B is negative and through the entire surface it is zero.

Answer: 3