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
$(a)$ Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by
$\left( E _{2}- E _{1}\right) \cdot \hat{ n }=\frac{\sigma}{\varepsilon_{0}}$
where $\hat{ n }$ is a unit vector normal to the surface at a point and $\sigma$ is the surface charge density at that point. (The direction of $\hat { n }$ is from side $1$ to side $2 .$ ) Hence, show that just outside a conductor, the electric field is $\sigma \hat{ n } / \varepsilon_{0}$
$(b)$ Show that the tangential component of electrostatic field is continuous from one side of a charged surface to another.
$(a)$ Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by
$\left( E _{2}- E _{1}\right) \cdot \hat{ n }=\frac{\sigma}{\varepsilon_{0}}$
where $\hat{ n }$ is a unit vector normal to the surface at a point and $\sigma$ is the surface charge density at that point. (The direction of $\hat { n }$ is from side $1$ to side $2 .$ ) Hence, show that just outside a conductor, the electric field is $\sigma \hat{ n } / \varepsilon_{0}$
$(b)$ Show that the tangential component of electrostatic field is continuous from one side of a charged surface to another.
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.
An infinite line charge produces a field of $9 \times 10^4 \;N/C$ at a distance of $2\; cm$. Calculate the linear charge density in $\mu C / m$
An infinite line charge produces a field of $9 \times 10^4 \;N/C$ at a distance of $2\; cm$. Calculate the linear charge density in $\mu C / m$
A uniform rod $AB$ of mass $m$ and length $l$ is hinged at its mid point $C$ . The left half $(AC)$ of the rod has linear charge density $-\lambda $ and the right half $(CB)$ has $+\lambda $ where $\lambda $ is constant . A large non conducting sheet of unirorm surface charge density $\sigma $ is also .present near the rod. Initially the rod is kept perpendicular to the sheet. The end $A$ of the rod is initially at a distance $d$ . Now the rod is rotated by a small angle in the plane of the paper and released. The time period of small angular oscillations is
A uniform rod $AB$ of mass $m$ and length $l$ is hinged at its mid point $C$ . The left half $(AC)$ of the rod has linear charge density $-\lambda $ and the right half $(CB)$ has $+\lambda $ where $\lambda $ is constant . A large non conducting sheet of unirorm surface charge density $\sigma $ is also .present near the rod. Initially the rod is kept perpendicular to the sheet. The end $A$ of the rod is initially at a distance $d$ . Now the rod is rotated by a small angle in the plane of the paper and released. The time period of small angular oscillations is
There is a solid sphere of radius $‘R’$ having uniformly distributed charge throughout it. What is the relation between electric field $‘E’$ and distance $‘r’$ from the centre ( $r$ is less than R ) ?
There is a solid sphere of radius $‘R’$ having uniformly distributed charge throughout it. What is the relation between electric field $‘E’$ and distance $‘r’$ from the centre ( $r$ is less than R ) ?