Name: Dielectrics in a Capacitor
Uploaded: 2019-07-19
Duration: 6 min 3 s
Description: Let us use our knowledge about dielectrics and capacitors to find out how the use of dielectrics can increase the capacitance of a system. Learn through this...

Question

Two thin parallel threads carry a uniform charge with linear densities $λ$ and $- λ$ . The distance between the threads is equal to $l$ . Find the potential of the electric field and the magnitude of its strength vector at the distance $r ≫ l$ , at an angle $θ$ to the vector $\vec{l}$ .

Accepted Answer

Two thin parallel threads carrying a uniform charge with linear densities $λ$ and $- λ$ are shown in the figure. The distance between the threads is $l$ .

Let point $θ$ be far apart to the threads, at a distance $\vec{l}$ , and at an angle $θ$ to the vector $\vec{l}$ (as shown in figure).

Thus, at point $P$ ,

$\vec{E} = {\vec{E}}_{+ λ} + {\vec{E}}_{- λ}$

$= \frac{λ}{2 π ε_{0}} \frac{\vec{r} + \frac{\vec{l}}{2}}{{|\vec{r} + \frac{\vec{l}}{2}|}^{2}} - \frac{λ}{2 π ε_{0}} \frac{\vec{r} - \frac{\vec{l}}{2}}{{|\vec{r} - \frac{l}{2}|}^{2}}$

$= \frac{λ}{2 π ε_{0}} [\frac{\vec{r} + \frac{\vec{l}}{2}}{r^{2} + \frac{l^{2}}{4} + r l cosθ} - \frac{\vec{r} - \frac{\vec{l}}{2}}{r^{2} + \frac{l^{2}}{4} - r l cosθ}]$

$= \frac{λ}{2 π ε_{0}} (\frac{\vec{l}}{r^{2}} - \frac{2 l \vec{r}}{r^{3}} cosθ)$

Hence, the magnitude of electric field vector,

$E = | \vec{E} | = \frac{λ l}{2 π ε_{0} r^{2}}, r > > l$

Also, potential at point $P$ ,

$φ = φ_{+ λ} + φ_{- λ} = \frac{λ}{2 π ε_{0}} \ln |\vec{r} + \frac{\vec{l}}{2}| - \frac{λ}{2 π ε_{0}} \ln |\vec{r} - \frac{\vec{l}}{2}|$

$= \frac{λ}{4 π ε_{0}} \ln \frac{r^{2} + r l cosθ + \frac{l^{2}}{4}}{r^{2} - r l cosθ + \frac{l^{2}}{4}} = \frac{λ l cosθ}{2 π ε_{0} r}, r > > l$

Two thin parallel threads carry a uniform charge with linear densities $λ$ and $- λ$ . The distance between the threads is equal to $l$ . Find the potential of the electric field and the magnitude of its strength vector at the distance $r ≫ l$ , at an angle $θ$ to the vector $\vec{l}$ .

Important Questions on ELECTRODYNAMICS

Learn and Prepare for any exam you want !

Two thin parallel threads carry a uniform charge with linear densities λ and -λ. The distance between the threads is equal to l. Find the potential of the electric field and the magnitude of its strength vector at the distance r≫l, at an angle θ to the vector l→.

Important Questions on ELECTRODYNAMICS

Learn and Prepare for any exam you want !

Two thin parallel threads carry a uniform charge with linear densities $λ$ and $- λ$ . The distance between the threads is equal to $l$ . Find the potential of the electric field and the magnitude of its strength vector at the distance $r ≫ l$ , at an angle $θ$ to the vector $\vec{l}$ .