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# Two infinitely long line charges each of linear charge density $\lambda$ are placed at an angle $\theta$ as shown in the figure. Find out electric field intensity at a point $P$ , which is at a distance $x$ from point $O$ along the angle bisector of line charges.

Last updated date: 29th Feb 2024
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Hint: The basic physical property of matter that causes it to experience a force when kept in an electric or magnetic field is electrical charge. An electric charge is correlated with an electric field and a magnetic field is generated by the moving electric charge. The electromagnetic field is recognized as a combination of electrical and magnetic fields.

Formula Used: We will use the following formula:
$E = \dfrac{{2k\lambda }}{r}$
Where
$E$ is the electrical charge
$\lambda$ is the linear charge density
$r$ is perpendicular distance of the point from the line charge
$k$ is the Coulomb’s constant.

Complete step by step solution:
Let us consider an infinitely long line charge whose linear charge density is $\lambda$
So, the magnitude of the electric field at any point at a distance of $r$ units will be
$E = \dfrac{{2k\lambda }}{r}$
And the direction of this electric field will be away from the wire

According to the question, it is given that $OP = x$
Let us assume that the length $AP = PB = r$
Now according to the figure,
$x\sin \dfrac{\theta }{2} = r$
Now we know that
$E = \dfrac{{2k\lambda }}{r}$
Now, we will replace the value of $r$ from the above expression. So, we will get
$E = \dfrac{{2k\lambda }}{{x\sin \dfrac{\theta }{2}}}$
So, the net electrical charge will be
${E_{net}} = 2E\sin \dfrac{\theta }{2}$
$= 2\left( {\dfrac{{2k\lambda }}{{x\sin \dfrac{\theta }{2}}}} \right) \times \sin \dfrac{\theta }{2}$
Therefore, the magnitude of the net electrical charge will be
${E_{net}} = \dfrac{{4k\lambda }}{x}$
And the direction of this net charge will be along $OP$ .

Note:
An electrical charge is a scalar quantity. In addition to having a magnitude and direction, the laws of vector addition such as triangle law of vector addition and parallelogram law of vector addition should also obey a quantity to be called a vector, only then is the quantity said to be a quantity of vector. When two currents meet at a junction, in the case of an electric current, the resulting current will be an algebraic sum and not the sum of the vector. Therefore, a scalar quantity is an electric current, although it has magnitude and direction.