Obtain the expression of electric field by a straight wire of infinite length and with linear charge density $'\lambda '$.
Obtain the expression of electric field by a straight wire of infinite length and with linear charge density $'\lambda '$.
Consider an infinitely long thin straight wire with uniform linear charge density $\lambda$.
Suppose we take the radial vector from $\mathrm{O}$ to $\mathrm{P}$ and rotate it around the wire. The points $\mathrm{P}, \mathrm{P}^{\prime}$,
$\mathrm{P}^{\prime \prime}$ so obtained are completely equivalent with respect to the charged wire.
This implies that the electric field must have the same magnitude at these points.
The direction of electric field at every point must be radial (outward if $\lambda>0$, inward $\lambda<0$ ).
Since the wire is infinite, electric field does not depend on the position of $\mathrm{P}$ along the length of
the wire.
The electric field is everywhere radial in the plane cutting the wire normally and its magnitude
depends only on the radial distance $r .$
Imagine a cylindrical Gaussian surface as shown in figure.
Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface
is zero.
At the cylindrical part of the surface $\overrightarrow{\mathrm{E}}$ is normal to the surface at every point and its magnitude
is constant since it depends only on $r .$
The surface area of the curved part is $2 \pi r l$, where $l$ is the length of the cylinder.
Flux through the Gaussian surface,
$=$ flux through the curved cylindrical part of the surface
$=E \times 2 \pi r l$
Similar Questions
Consider a sphere of radius $R$ with charge density distributed as :
$\rho(r) =k r$, $r \leq R $
$=0$ for $r> R$.
$(a)$ Find the electric field at all points $r$.
$(b)$ Suppose the total charge on the sphere is $2e$ where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.
Consider a sphere of radius $R$ with charge density distributed as :
$\rho(r) =k r$, $r \leq R $
$=0$ for $r> R$.
$(a)$ Find the electric field at all points $r$.
$(b)$ Suppose the total charge on the sphere is $2e$ where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.
Two long thin charged rods with charge density $\lambda$ each are placed parallel to each other at a distance $d$ apart. The force per unit length exerted on one rod by the other will be $\left(\right.$ where $\left.k=\frac{1}{4 \pi \varepsilon_0}\right)$
Two long thin charged rods with charge density $\lambda$ each are placed parallel to each other at a distance $d$ apart. The force per unit length exerted on one rod by the other will be $\left(\right.$ where $\left.k=\frac{1}{4 \pi \varepsilon_0}\right)$
If the total charge enclosed by a surface is zero, does it imply that the electric field everywhere on the surface is zero ? Conversely, if the electric field everywhere on a surface is zero, does it imply that net charge inside is zero.
If the total charge enclosed by a surface is zero, does it imply that the electric field everywhere on the surface is zero ? Conversely, if the electric field everywhere on a surface is zero, does it imply that net charge inside is zero.
Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude $17.0\times 10^{-22}\; C/m^2$. What is $E$:
$(a)$ in the outer region of the first plate,
$(b)$ in the outer region of the second plate, and
$(c)$ between the plates?
Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude $17.0\times 10^{-22}\; C/m^2$. What is $E$:
$(a)$ in the outer region of the first plate,
$(b)$ in the outer region of the second plate, and
$(c)$ between the plates?
Find the force experienced by the semicircular rod charged with a charge $q$, placed as shown in figure. Radius of the wire is $R$ and the line of charge with linear charge density $\lambda $ is passing through its centre and perpendicular to the plane of wire.
Find the force experienced by the semicircular rod charged with a charge $q$, placed as shown in figure. Radius of the wire is $R$ and the line of charge with linear charge density $\lambda $ is passing through its centre and perpendicular to the plane of wire.