Question

Suppose electrons move through a copper wire at speed v. Call the linear charge densities in the Earth reference frame λ_Ep for the fixed positive ions in the wire and λ_En for the (negative) electrons. Observer E in the Earth reference frame (which is also the reference frame of the positive ions) measures the wire to be electrically neutral. Observer M is moving along with the electrons (in the same direction as the electrons and at speed v).

A) Calculate the linear charge density of the wire as measured by M in terms of λ_En if the speed is 4.5 ×105m/s. λ_M / λ_En = ?

B) Calculate the linear charge density of the wire as measured by M in terms of λ_En if the speed is 7.5 mm/s (an achievable speed for electrons in copper wire). λ_M / λ_En = ?

Answer 1

(A) Linear charge density on the wire is, E_p + E_n = 0 E_p = -E_n from the fixed observer E, now observer M is moving with electrons hence will see the negative charge density as, E�_n = E_n positive charge density written as, E�_p = E_p times 1/squareroot (1-v^2/c^2) = E_p times squareroot (1-c^2-v^2/c^2) = E_p times c/squareroot c^2 - v^2 therefore the net charge density will be E�_p + E�_n = -E_nc/squareroot c^2 - v^2 + E_n, (E_p = -E_n) = squareroot c^2 - v^2 - c/squareroot c^2 - v^2 E_n = squareroot 1-v^2/c^2 - 1/squareroot 1-v^2/c^2 E_n when, v = 4.5 times 10^5 v/c = 4.5 times 10^5/3 times 10^8 = 1.5 times 10^-3 E�_p + E�_n = squareroot 1-(1.5 times 10^-3) ^2 -1/squareroot 1-(1.5 times 10^-3) ^2 E_n = -1.12^E_n (B) when, v = 7.5 times 106-3 v/c = 7.5 times 10^-3/3 times 10^8 = 2.5 times 10^-11 E�_p + E�_n = squareroot 1-(2.5 times 10^-11) ^2-1/squareroot 1-(2.5 times 10^-11) ^2 E_n = -1E_n