Question
Get the electric field due to on infinitely long straight uniformly changed wire.
✓
Answer
$►$ Consider an infinitely long thin straight wire with uniform linear charge density $\lambda$.
$►$ Suppose we take the radial vector from $O$ to $P$ and rotate it around the wire. The points $P , P ^{\prime}$, $P ^{\prime \prime}$ 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 if $\lambda<0 )$.
$►$ Consider a pair of line elements $P_1$ and $P_2$ of the wire, as shown in Fig. $(a)$
$►$ The electric fields produced by the two elements of the pair when summed, give a resultant electric field which is radial $($the components normal to the radial vector cancel$)$.
$►$ This is true for any such pair and hence the total field at any point $P$ is radial.
$►$ The electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance $r$.
$►$ To calculate the field, imagine a cylindrical Gaussian surface, as shown in the Fig. $(b)$.
$►$ Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero.
$\phi=$ Flux through the curved surface of the cylinder
$\therefore \phi=\overrightarrow{ E } \cdot \overrightarrow{ S }$
$\therefore \phi= ES \cos 0 \quad(\overrightarrow{ E } \| \overrightarrow{ S })$
$\therefore \phi= E (2 \pi r l) \text { (S-area of curved surface) } \ldots$
$►$ As per Gauss's law, $\phi=\frac{q}{\varepsilon_0}$
but $q=\lambda l ($electric charge surrounded by Gaussian surface$)$
$\therefore \phi=\frac{\lambda l}{\varepsilon_0}$
$►$ Comparing equation $(1)$ and equation $(2),$
$E (2 \pi r l)=\frac{\lambda l}{\varepsilon_0}$
$E =\frac{1}{2 \pi \varepsilon_0} \cdot \frac{\lambda}{r}$
$►$ Electric field in vector form,
$\overrightarrow{ E }=\frac{1}{2 \pi \varepsilon_0} \cdot \frac{\lambda}{r} \hat{n}$
Where, $\hat{n}$ is unit vector in the direction of electric field.
OR
$\overrightarrow{ E }=\frac{2 k \lambda}{r} \hat{n}\left[\text { where } k=\frac{1}{4 \pi \varepsilon_o}\right]$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The sunlight reaching Earth has maximum electric field of $810Vm^{-1}$. What is the maximum magnetic field in this light?
→When $1.0 \times 10^{12}$ electrons are transferred from one conductor to another, a potential difference of 10V appears between the conductors. Calculate the capacitance of the two-conductor system.
→What is space wave propagation? State the factors which limit its range of propagation. Derive an expression for the maximum line of sight distance between two antennas for space wave propagation.
A candle flame 1.6cm high is imaged in a ball bearing of diameter 0.4cm. If the ball bearing is 20cm away from the flame, find the location and the height of the image.
Figure shows a light rod of length l rigidly attached to a small heavy block at one end and a hook at the other end. The system is released from rest with the rod in a horizontal position. There is a fixed smooth ring at a depth h below the initial position of the hook and the hook gets into the ring as it reaches there. What should be the minimum value of h so that the block moves in a complete circle about the ring?
- Derive the relation between the decay constant and half life of a radioactive substance.
- A radioactive element reduces to 25% of its initial mass in 1000 years. Find its half life.
A neutron is absorbed by a $^{6}_{3}\text{Li}$ nucleus with the subsequent emission of an alpha particle.
→- Calculate the energy released, in MeV, in this reaction.
- Write the corresponding nuclear reaction.
Explain with the help of a labelled diagram the underlying principle and working of a step-up transformer. Why cannot such a device be used to step-up d.c. voltage?
A charge Q is distributed uniformly over a metallic sphere of radius R. Obtain the expressions for the electric field (E) and electric potential (V) at a point 0 < x < R.
Show on a plot the variation of E and V with x for 0 < x < 2R.
Show on a plot the variation of E and V with x for 0 < x < 2R.
Define the terms threshold frequency and stopping potential in relation to the phenomenon of photoelectric effect. How is the photoelectric current affected on increasing the (1) frequency (2) intensity of the incident radiations and why?