Integrated Concepts The practical limit to an electric field in air is about 3.00 6? × 10? N/C. Above this strength, sparking takes place because air begins to ionize and charges flow, reducing the field. (a) Calculate the distance a free proton must travel in this field to reach 3.00% of the speed of light, starting from rest. (b) Is this practical in air, or must it occur in a vacuum?

Solution : In the above question they have given the limit to an electric field in air is about 3.00 × 10 N/C. Step 1 (a) Calculate the distance a free proton must travel in this field to reach 3.00% of the speed of light, starting from rest First we need to calculate the distance of a free proton must travel in this field to reach 3.00% of the speed of light, So we need to calculate the coulomb’s law of electrostatic field is,’ F = qE Here q s charge of proton, E is the electric field. Next we need to calculate from Newton’s second law F = ma Here, m is the mass of proton and a is the acceleration of proton. Step 2 In the above two,Equate Newton’s law and coulomb’s electrostatic field is, ma = qE a = qE/m Step 3 We need to find the distance travelled by proton to reach 3% of the speed of light, Calculate from Newton’s equation of motion, 2 2 v = vo+ 2ax Where, v is the final velocity of electron, vo s the initial velocity of electron, x is the distance travelled by electron, a is the acceleration of electron. We now need to substitute the 0 for v we get, v = v + 2ax o 0 = v + 2ax o In the above term the 0 is no value so we taking x in form of we get as, x = v / 2a Then substitute qE/m for a we get as, 2 x = v m / 2qE