3 3 is applied aWhat is the magnitude and direction of the dipole moment bWhat

3 3 is applied awhat is the magnitude and direction

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??? = (3?̂ + 3?̂)?/? is applied. (a) What is the magnitude and direction of the dipole moment? (b) What is the magnitude and direction of the torque on the dipole? (c) Do the electric fields of the charges q 1 and q 2 contribute to the torque on the dipole? Briefly explain your answer.
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17 Chapter 2 Electric Potential 2.1. Potential and Potential Energy In the introductory mechanics course, we have seen that gravitational force from the Earth on a particle of mass m located at a distance r from Earth’s center has an inverse - square form: ? ? = −? ?? ? 2 (2.1.1) Where G 6.67 10 11 N m 2 /kg 2 is the gravitational constant and is a unit vector pointing radially outward. The Earth is assumed to be a uniform sphere of mass M . The corresponding gravitational field ? , defined as the gravitational force per unit mass, is given by ? = ? 𝑔 ? = − ?? ? 2 (2.1.2) Notice that g only depends on M , the mass which creates the field, and r , the distance from M . Figure 2.1.1 Consider moving a particle of mass m under the influence of gravity (Figure 2.1.1). The work done by gravity in moving m from A to B is ? ? = ∫ ? ? ∙ ? ? = ∫ ( ??? ? 2 ) ?? = [ ??? ? ] ? ? ? ? = ??? ( 1 ? ? 1 ? ? ) ? ? ? ? 
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18 The result shows that ? ? is independent of the path taken; it depends only on the endpoints A and B . It is important to draw distinction between ? ? , the work done by the field and ? ??? , the work done by an external agent such as you. They simply differ by a negative sign: ? ? = − ? ??? . Near Earth’s surface, the gravitational field ? is approximately constant, with a magnitude ? = ?? ? 𝐸 2 9.81 ?/? 2 , where ? ? is the radius of Earth. The work done by gravity in moving an object from height y A to y B (Figure 2.1.2) is ? ? = ∫ ? ? ∙ ? ? = ∫ ?????𝜃?? = − ∫ ?????∅?? = − ? ? ? ? ???? = −??( ? ? ? ? ? ? − ? ? ) (3.1.4) Figure 2.1.2 Moving a mass m from A to B . The result again is independent of the path, and is only a function of the change in vertical height y B y A . In the examples above, if the path forms a closed loop, so that the object moves around and then returns to where it starts off, the net work done by the gravitational field would be zero, and we say that the gravitational force is conservative. More generally, a force ? ⃗⃗ is said to be conservative if its line integral around a closed loop vanishes: ? ⃗⃗ ∙ ?? = 0 (2.1.5) When dealing with a conservative force, it is often convenient to introduce the concept of potential energy U . The change in potential energy associated with a conservative force ? ⃗⃗ acting on an object as it moves from A to B is defined as: ∆? = ? ? − ? ? = ∫ ? ∙ ?? ? ? = −? (2.1.6) where W is the work done by the force on the object. In the case of gravity, ? = ? ? and from Eq.(2.1.3), the potential energy can be written as
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19 ? ? = − ??? ? + ? 0 (2.1.7) where U 0 is an arbitrary constant which depends on a reference point. It is often convenient to choose a reference point where U 0 is equal to zero. In the gravitational case, we choose infinity to be the reference point, with U 0 ( r ) 0. Since U g depends on the reference point chosen, it is only the potential energy difference U g that has physical importance. Near Earth’s surface where the gravitational field ? is approximately constant, as an object moves from the ground to a height h , the change in potential energy is U g mgh
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