Two protons separated by a distance 'r' travelling n the same direction with velocity 'v' show that the ratio of magnetic and electric forces acting between them is (v/c)^2, c is the speed of light.

Asked by Tanya raj | 21st Aug, 2015, 08:19: PM

Expert Answer:

The electric force between the two protons is given as
 
begin mathsize 14px style straight F subscript straight E equals fraction numerator 1 over denominator 4 πε subscript 0 end fraction straight q squared over straight r squared end style
Now, the magnetic force is given as FB = qvB
 
To find B:
 
The magnetic field is given from Biot-Savarts law as
 
begin mathsize 14px style straight B equals fraction numerator straight mu subscript 0 over denominator 4 straight pi end fraction IdL over straight r squared end style
dL is the length element of the conductor. However, we do not have any conductor here, but only protons moving.
 
So, the expression will be altered for charge and not current.
 
We know that I = q/t, so in time dt, the current moves a length dL. So, we get
 
I dL = q dL/dt = qv
 
Hence, the magnetic field is
 
begin mathsize 14px style straight B equals fraction numerator straight mu subscript 0 over denominator 4 straight pi end fraction qv over straight r squared end style
 
Therefore, the magnetic force is given as
 
begin mathsize 14px style straight F subscript straight B equals qvB equals fraction numerator straight mu subscript 0 over denominator 4 straight pi end fraction qv over straight r squared qv straight F subscript straight B equals fraction numerator straight mu subscript 0 over denominator 4 straight pi end fraction fraction numerator straight q squared straight v squared over denominator straight r squared end fraction end style
 
Hence, the ratio of electric and magnetic forces is
 
begin mathsize 14px style straight F subscript straight B over straight F subscript straight E equals fraction numerator fraction numerator straight mu subscript 0 straight q squared straight v squared over denominator 4 πr squared end fraction over denominator fraction numerator straight q squared over denominator 4 πε subscript 0 straight r squared end fraction end fraction equals straight mu subscript 0 straight epsilon subscript 0 straight v squared end style
 
Now, we know that the speed of light is given as
 
begin mathsize 14px style straight c equals square root of fraction numerator 1 over denominator straight mu subscript 0 straight epsilon subscript 0 end fraction end root therefore straight c squared equals fraction numerator 1 over denominator straight mu subscript 0 straight epsilon subscript 0 end fraction therefore straight mu subscript 0 straight epsilon subscript 0 equals 1 over straight c squared end style
 
Therefore, we get the ratio as
 
begin mathsize 14px style straight F subscript straight B over straight F subscript straight E equals straight mu subscript 0 straight epsilon subscript 0 straight v squared equals straight v squared over straight c squared end style

Answered by Romal Bhansali | 22nd Aug, 2015, 09:05: PM