HW1 solutions

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1 Homework 1 Solutions 1. Sphere #1 exerts electric force on Sphere #2. Both spheres are uniformly charged. How does the magnitude of the force change if the absolute value of the charge of #1 is decreased and the magnitude of the velocity of #1 is decreased? Assume that all other parameters remain fixed. a) It increases *b) It decreases c) It does not change d) There is not enough information to determine - The magnitude of the force is given by Coulomb's Law, F12=k*|q1|*|q2|/r^2, where k is the Coulomb constant (9.0E9), |q1| and |q2| are the absolute values of the charges, and r is the distance between the charges. - According to this relationship decreasing the absolute value of the charge of #1 will decrease the magnitude of the force. - Since the magnitude of the velocity of #1 does not appear in this formula, this means that the magnitude of the force does not depend on this variable. - Here, since one change causes a decrease and the other change has no effect, overall we can conclude there will be a decrease. -------------------------------------------------------------- 2. A -1.63 C point charge #1 is located at (0,0) and a -5.85 C point charge #2 is located at (9.1,6.1). What is the direction of the force in Newtons exerted by #2 on #1 expressed as an angle between 0 and 360 degrees measured counterclockwise from the +x direction? The coordinates are given in meters. *a) 214 b) 330 c) 19 d) 127 e) 63 Solution: - The force exerted by #2 on #1 is given by Coulomb's Law: F21=(k*q2*q1/r^2)*R21, where k is the Coulomb constant (9.0E9), q1 and q2 are the charges, r is the distance between them, and R21 is the unit vector pointing from #2 to #1. - To find the angle theta of the R21 unit vector, since q1 is located at (0,0) and q2 is located at (x,y) we have tan(theta)=(0-y)/(0-x)=y/x so theta=arctan(y/x). The minus signs which cancel out are indicated because R21 points the opposite direction of R12. - It is important to apply trigonometry rules for calculating arctan. If the vector points toward quadrant I you can directly calculate arctan(y/x) with a calculator. If it points toward quadrant II or towards quadrant III you must add 180 to the calculator result. If it points towards quadrant IV then you must add 360 to the calculator result. Also, be sure your calculator is in degrees mode (not radians mode)! - Here, based on the coordinates of the charges, R21 points towards quadrant III (180 to 270 degrees) so we get theta=arctan(6.1/9.1)=214 degrees.

2 - The final step is to consider that F21=(k*q1*q2/r^2)*R21 implies that the direction of the force depends on the signs of q1 and q2 since the unit vector R21 is multiplied by q1*q2. If q1*q2 is positive then the force points in the same direction as R21, but if q1*q2 is negative it points in the opposite direction, meaning that the angle of the force would be theta plus or minus 180 degrees (whichever gives an angle between 0 and 360, since the question asks for this). - Here q1*q2 is positive, so we conclude that the angle of the force is 214 degrees -------------------------------------------------------------- 3. A 6.31 C point charge #1 is located at (0,0) and a 9.55 C point charge #2 is located at (-7.3,0.17). What is the absolute value of the x component of the force in Newtons exerted by #1 on #2? The coordinates are given in meters. Since this is asking for the absolute value, give a positive number. The next question will ask what the sign is. a) 4.62E+10 *b) 1.02E+10 c) 1.31E+10 d) 3.16E+10 e) 5.95E+10 4. For the previous problem (#3. ) is the value of the quantity positive or negative? a) positive *b) negative Solution: - The force exerted by #1 on #2 is given by Coulomb's Law: F12=(k*q1*q2/r^2)*R12, where k is the Coulomb constant (9.0E9), q1 and q2 are the charges, r is the distance between them, and R12 is the unit vector pointing from #1 to #2 - The x component of the force is F12x=(k*q1*q2/r^2)*cos(theta), where theta is the angle of the unit vector (measured counterclockwise from the +x direction). - Since q1 is located at (0,0) and q2 is located at (x,y) we have r=sqrt(x^2+y^2)=sqrt((-7.3)^2+(0.17)^2)=7.3 and cos(theta)=x/r. - Plugging in the numbers we get F12x=((9.00E+09)*(6.31)*(9.55)/(7.3)^2)*(-7.3/7.3)=-1.02E+10. -------------------------------------------------------------- 5. If both point charges are positive the force that #1 exerts on #2 is in the direction pointing from #1 to #2 *a) True b) False Solution: This is true because it is implied by the full vector form of Coulomb's Law which was discussed in lecture #3 -------------------------------------------------------------- 6. Suppose a point charge #2 with charge -0.479 C is located at (0,0) and a point charge #1 with charge 0.69 C is located at (1.41,1.14). Which of the following is true about the components of the force exerted by #1 on #2?

3 -------------------------------------------------------------- 7. Two objects are small enough to be considered point charges and #1 exerts electric force on #2. If the number of electrons in #2 is 67, distance between #1 and #2 is 5.30E-05 meters, charge of #1 is 0.026 Coulombs, and number of protons in #2 is 177, what is the magnitude of the force in Newtons? *a) 1.47 b) 3.95 c) 5.68 d) 8.17 e) 4.43 Solution: - Note that since all the quantities are given in SI units we can calculate the answer in SI units without needing to do any unit conversions. - The magnitude of the force is given by Coulomb's Law: F=k*|q1|*|q2|/r^2, where k is the Coulomb constant (9.0E9 in SI units), q1 is the charge of #1, q2 is the charge of #2, and r is distance between #1 and #2. - The charge of #2 is given by q2=e*(Np-Ne), where e is the proton charge (1.60E-19 Coulombs), Np is the number of protons in #2, and Ne is number of electrons in #2.

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4 - Plugging in the numbers gives q2=1.60E-19*(177-67)=1.76E-17 - Going back to F=k*|q1|*|q2|/r^2 and plugging in the numbers gives F=9.00E+09*|0.026|*|1.76E-17|/5.30E- 05^2=1.47 -------------------------------------------------------------- 8. Point charge #1 exerts electric force on point charge #2. If the distance between the two charges is increased 30x then by what number would the absolute value of the charge of #1 have to be multiplied to cause the magnitude of the force to be increased three-fold? Assume all other parameters remain unchanged. a) A number >=1.00E+03 and <1.60E+03 *b) A number >=1.60E+03 and <3.00E+03 c) A number >=3.00E+03 and <5.00E+03 d) A number >=5.00E+03 and <9.90E+03 e) None of the above Solution: - The magnitude of the force is given by Coulomb's Law: F=k*|q1|*|q2|/r^2, where k is the Coulomb constant, |q1| the absolute value of the charge of #1, |q2| the absolute value of the charge of #2, and r the distance between them. - To find what factor |q1| would have to be multiplied by solve for it to find |q1|=F*r^2/(k*|q2|) - To simplify the analysis, suppose the initial values are F=r=|q2|=1, so initially |q1|=1/k - Then, after the specified changes F=3, r=30, and |q2|=1 - Plugging in the chnaged values we find |q1|=F*r^2/(k*|q2|)=3*30^2/(k*1)=2.70E+03*(1/k) - Comparing to the initial |q1|=1/k we see that |q1| must be multiplied by 2.70E+03 -------------------------------------------------------------- 9. In considering the total electric forces on biomolecules dissolved in water the presence of the water has a negligible effect a) True *b) False Solution: This is false because contradicts facts discussed in lecture #3. The dielectric constant of water has a strong influence on total electric forces due to the forces exerted by polarized water molecules on the biomolecules. -------------------------------------------------------------- 10. An uncharged metal object can be charged by induction, by bringing a charged object near the metal object and then connecting the metal object to ground *a) True b) False Solution: This is true because it describes the charging by induction discussed in lecture #2 --------------------------------------------------------------

5 11. Point charge #2 experiences a certain electric force due to point charge #1. Suppose the following changes are made: the separation between the two charges is increased by a factor of ten and the absolute value of the charge of #2 is decreased 100x. This would cause magnitude of the force to be multiplied by what factor? Assume all other parameters are fixed. a) A number >=1.00E-03 and <0.0015 b) A number >=0.0015 and <0.003 c) A number >=0.003 and <0.005 d) A number >=0.005 and <0.0099 *e) None of the above Solution: - The magnitude of the force is given by Coulomb's Law: F=k*|q1|*|q2|/r^2, where k is the Coulomb constant, |q1| the absolute value of the charge of #1, |q2| the absolute value of the charge of #2, and r the distance between them. - To simplify the analysis suppose the initial values are |q1|=|q2|=r=1, so initially F=k - Then, after the specified changes |q1|=1, |q2|=0.01, r=10 - Plugging in the changed values we find F=k*|q1|*|q2|/r^2=k*1*0.01/10^2=1.00E-04*(k) - Comparing to the initial F=k we see that F has been multiplied by 1.00E-04 -------------------------------------------------------------- 12. Body #1 exerts electric force on body #2. Suppose both are small enough to be considered point charges. What would the number of protons in #2 have to be if the distance between #1 and #2 is 7.30E-05 meters, number of electrons in #2 is 91, magnitude of the force is 6.82 Newtons, and charge of #1 is 0.031 Coulombs? Assume the charge of #2 is positive. a) 367 b) 789 *c) 905 d) 221 e) 115 Solution: - Note that since all the quantities are given in SI units we can calculate the answer in SI units without needing to do any unit conversions. - The magnitude of the force is given by F=k*|q1|*|q2|/r^2, where k is the Coulomb constant (9.0E9 in SI units), q1 is the charge of #1, q2 is the charge of #2, and r is distance between #1 and #2. - The charge of #2 is given by q2=e*(Np-Ne), where e is the proton charge (1.60E-19 Coulombs), Np is the number of protons in #2, and Ne is number of electrons in #2. - To find what Np must be, solve for it to get Np=(q2 + Ne*e)/e. - q2 is is specified to be positive and found by solving F=k*|q1|*|q2|/r^2 which gives q2=(F*r^2)/(k*|q1|)=(6.82*7.30E-05^2)/(9.00E+09*|0.031|)=1.30E-16

6 - Going back to Np=(q2 + Ne*e)/e and plugging in the numbers gives Np=(1.30E-16 + 91*1.60E-19)/1.60E- 19=905 -------------------------------------------------------------- 13. Every electron has the same amount of negative charge and this cannot be changed *a) True b) False Solution: This is a fact discussed in lecture #1. -------------------------------------------------------------- 14. Electrical "ground" refers to a connection of an object to the Earth so that electrons can move to Earth from the object or vice versa *a) True b) False Solution: This is a fact discussed in lecture #2 -------------------------------------------------------------- 15. Contact forces, spring forces, and friction forces are all fundamentally due to electric force *a) True b) False Solution: This is a fact discussed in lecture #1 -------------------------------------------------------------- 16. Suppose a point charge #2 with charge -0.286 C is located at (0,0) and a point charge #1 with charge -0.698 C is located at (-0.72,-1.23). Which of the following is true about the components of the force exerted by #2 on #1? a) The x and y components are both positive b) The x component is positive and y component is negative c) The x component is negative and y component is positive *d) The x and y components are both negative e) There is not enough information to determine Solution: The positions of q1 and q2 are plotted below and since both charges have the same sign they repel each other, so the force on #1 is in the direction shown by the arrow (away from #2). Since it points partly to the left the x- component of the force is negative. Since it points partly downwards the y-component of the force is negative.

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7 -------------------------------------------------------------- 17. When an atom becomes electrically polarized the positions of the electrons shift relative to the nucleus *a) True b) False Solution: This is a fact discussed in lecture #3 -------------------------------------------------------------- 18. Two uniformly charged spheres exert electric force on each other. If the distance between the charges is decreased and the absolute value of the charge of #2 is decreased, how does the magnitude of the force change? Assume that all other parameters remain fixed. a) It increases b) It decreases c) It does not change *d) There is not enough information to determine - The magnitude of the force is given by Coulomb's Law, F12=k*|q1|*|q2|/r^2, where k is the Coulomb constant (9.0E9), |q1| and |q2| are the absolute values of the charges, and r is the distance between the charges. - According to this relationship decreasing the distance between the charges will increase the magnitude of the force. - Decreasing the absolute value of the charge of #2 will decrease the magnitude of the force. - Here, since one change causes an increase but the other change causes a decrease it is not possible to determine without additional information if there will overall be an increase, decrease, or no change. All three are possible. -------------------------------------------------------------- Points: 4 19. Three point charges exert electric forces on each other. Assume there are no other forces acting on these charges. A -9.8 C point charge #1 is located at (3.5,0), a -2 C point charge #2 is located at (-7.4,0) and a 0.23 C point charge #3 is located at (2.6,0). The coordinates are given in meters. What is the magnitude of the total force on #2? The next question will ask what is the direction of the force.

8 a) 8.744E+09 b) 9.651E+09 c) 6.759E+09 *d) 1.443E+09 e) 5.349E+09 Points: 1 20. For the previous problem (#19) what is the direction of the force? a) +x direction *b) -x direction Solution: - The force exerted by #1 on #2 is given by Coulomb's Law: F12=(k*q1*q2/r12^2)*R12, where k is the Coulomb constant (9.0E9), q1 and q2 are the charges, r12 is the distance between #1 and #2, and R12 is the unit vector pointing from #1 to #2 - All three charges have y coordinates equal to zero, which means they lie along the x axis, which implies that all force vectors will lie on the x axis. The forces will only have x components, not y components. - The x component of the force exerted by #1 on #2 is F12x=(k*q1*q2/r12^2)*cos(theta12), where theta12 is the angle of the unit vector pointing from #1 to #2. - Here cos(theta12)=delta(x)/r12=(x2-x1)/|x2-x1|=((-7.4)-(3.5))/|(-7.4)-(3.5)|=-1 - Plugging in the numbers we get F12x=((9.0E9)*(-9.8)*(-2)/(10.9)^2)*(-1)=-1.485E+09 - The x component of the force exerted by #3 on #2 is F32x=(k*q3*q2/r32^2)*cos(theta32), where theta32 is the angle of the unit vector pointing from #3 to #2. - Here cos(theta32)= (x2-x3)/r32=(x2-x3)/|x2-x3|=((-7.4)-(2.6))/|(-7.4)-(2.6)|=-1 - Plugging in the numbers we get F32x=((9.0E9)*(0.23)*(-2)/(10)^2)*(-1)=4.140E+07 - The x component of the total force on #2 is equal to the x component of the force exerted by #1 on #2 plus the x component of the force exerted by #3 on #2: Fon2x=F12x + F32x=(-1.485E+09)+(4.140E+07)=-1.443E+09 - Since there is only an x component and no y component, the magnitude of the force is Fon2=|Fon2x|=1.443E+09 - That the x component of the force is negative means the force points in the -x direction

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