Define the linear charge density of an infinitely long thread, if the work force field to move a charge Q = 1 nC at a distance r ₁ = 5 cm to r ₂ = 2 cm in the direction perpendicular to the filament is 50 uJ.

Определить линейную плотность заряда бесконечно длинной нити, если работа сил поля по перемещению заряда Q = 1 нКл с расстояния r₁ = 5 cм до r₂ = 2 см в направлении перпендикулярном нити равна 50 мкДж.

I am not sure how much your book gives you.

I will assume that you know that:
E = [ 1/(2pieo) ] lambda/r
where lambda = charge per unit length
and
r is distance perpendicular to the line
then
Voltage potential of point a at ra with respect to potential at point b at rb is:
Va -Vb = integral from a to b of E dr
= [lambda/(2pieo)]integral dr/r
= [lambda/(2pieo)] ln (rb/ra)
calculate that
now
work done = charge* voltage difference

To determine the linear charge density of an infinitely long thread, we can use the formula for electric potential energy:

ΔU = Q ∆V

Here, ΔU is the change in electric potential energy, Q is the charge being moved, and ∆V is the change in electric potential. In this case, the charge being moved is Q = 1 nC (1 nanoCoulomb), and the change in distance is from r₁ = 5 cm to r₂ = 2 cm.

First, let's convert the given values into SI units:

Q = 1 nC = 1 × 10^(-9) C
r₁ = 5 cm = 5 × 10^(-2) m
r₂ = 2 cm = 2 × 10^(-2) m

To find the change in electric potential ∆V, we use the formula:

∆V = k * Q / r

Where k is the electrostatic constant equal to approximately 8.99 × 10^9 Nm²/C².

Substituting the given values, we have:

∆V = (8.99 × 10^9 Nm²/C²) * (1 × 10^(-9) C) / (2 × 10^(-2) m) - (5 × 10^(-2) m)

∆V = (8.99 × 10^9 Nm²/C²) * (1 × 10^-9 C) / (2 × 10^(-2) m) - (5 × 10^(-2) m)

Simplifying the expression, we get:

∆V = 3.5976 × 10^3 Nm/C

Now, we can calculate the change in electric potential energy using the formula:

ΔU = Q * ∆V

Substituting the given values, we have:

50 uJ = (1 × 10^(-9) C) * (3.5976 × 10^3 Nm/C)

Converting Joules into microJoules, we have:

50 uJ = 5 × 10^(-5) J

Therefore, Q * ∆V = 5 × 10^(-5) J

Now, we can solve for Q:

Q = (5 × 10^(-5) J) / (3.5976 × 10^3 Nm/C)

Q ≈ 1.39 × 10^(-8) C

Finally, we can determine the linear charge density, λ, using the formula:

λ = Q / L

Since the thread is infinitely long, L is considered to be infinite. In this case, the linear charge density, λ, is constant throughout the thread. Therefore, λ is equal to:

λ = Q / L ≈ (1.39 × 10^(-8) C) / (∞)

As ∞ represents an infinitely long thread, the linear charge density λ is undefined.

Therefore, for an infinitely long thread, the linear charge density cannot be determined.