Two electrons, A and B, approach each other and follow symmetrical trajectories

Question

Two electrons, A and B, approach each other and follow symmetrical trajectories as shown below. At their closest separation, which coincides with their mutual crossing of the x axis, they are 2.1 times 10^10 m apart. (Look up properties of electrons, please). At the closest approach, determine the electrostatic force that acts on each electron and express each of these vectors in unit vector notation with correct signs. Repeat for the acceleration vector for each particle. Assuming this all occurs near Earth's surface, Earth's gravity must also act on the electrons. What percentage of the electrostatic force that acts on each electron is the gravitational force? Are we justified in ignoring the force of gravity? Briefly explain your reasoning. If instead A and B were both protons (look up their properties too) following the same trajectories, how would your answers to part a change, if at all. Explain your reasoning.

Explanation / Answer

Part a

At point of closest approach,

                Distance between electron (d) = 2.1 * 10^-10 m

Electrostatic force on right electron (F21)= Ke2 /d2 along +x axis

                                                                                = 9*10^9 * 1.6*1.6*10^-38 / 2.1*2.1*10^-20

                                                                                = 5.22449 * 10^-18 N

                                                                                F21 =( 5.22449 * 10^-18 )i N.

Acceleration of right electron (a21)=F21 /m =0.574 * 10^13 =5.74 * 10^12 m/s^2.

                                                                a21=(5.74 * 10^12)i m/s^2

Electrostatic force on left electron (F12)= Ke2 /d2 along -x axis

= 9*10^9 * 1.6*1.6*10^-38 / 2.1*2.1*10^-20

                                                                                = 5.22449 * 10^-18 N

                                                                                F12 = - ( 5.22449 * 10^-18 )i N.

Acceleration of left electron (a12)=F12 /m =0.574 * 10^13 =5.74 * 10^12 m/s^2.

a12=( - 5.74 * 10^12)i m/s^2.

Part b

Gravitational force on electron(Fg)= mg=9.1*10^-31 * 9.8 =89.18 * 10^-31 N

                Percentage=Fg /Fe *100 =17.06961 * 10^-11

As gravitational force is very very week in comparison to electrostatic force . so we can neglect force of gravity here.

Part c

Force on both protons will be same as on electrons ,as electrostatic force depends upon charge and distance between objects. Which are same in both cases,

Acceleration of protons will be different from electron, as their mass is different.

Acceleration of left proton (a12)=F12 /m =( 5.22449 * 10^-18 )/1.67 * 10^-27 = - (3.1284 * 10^9 )i m/s^2

Acceleration of right proton (a21)=F21 /m =( 5.22449 * 10^-18 )/1.67 * 10^-27 = (3.1284 * 10^9

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