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How do you determine the electric field near a charged conducting surface?
Suppose, a spherical conductor of radius r placed in a medium of permittivity ε has + Q amount of charge [Fig.]. The electric field is to be calculated at a point P close to the sphere.
If we place a unit positive charge at P, the force acting on it gives the electric field at P. It can be proved that, to determine electric field at an external point due to a charged spherical conductor, we can assume that the whole charge of the sphere is concentrated at the centre of the sphere. So the distance between the charge concentrated at the centre O of the sphere and the unit positive charge at P may be taken to be equal to the radius of the sphere.
Electric field at P
E = \(\frac{Q}{4 \pi \epsilon r^2}\) = \(\frac{Q}{4 \pi \kappa \epsilon_0 r^2}\)
[k = dielectric constant of the medium]
= \(\frac{4 \pi r^2 \sigma}{4 \pi \kappa \epsilon_0 r^2}\) [∵ Q = 4πr2σ]
∴ E = \(\frac{\sigma}{\kappa \epsilon_0}\)
In vector form, \(\vec{E}\) = \(\frac{\sigma}{\kappa \epsilon_0} \hat{n}\)
where \(\hat{n}\) is the unit vector normal in the direction of \(\vec{E}\) as shown in the figure.
It may be noted that the electric field E does not depend on the radius of the sphere. So the above equation is applicable not only to a spherical conductor, hut also to a conductor of any shape.
Electric Field Lines Or Electric Lines Of Force
Eiectric field intensity acts on every point of an electric field. This intensity has a definite magnitude and direction. But the magnitude and direction is different at different points. If a free isolated unit posiilve charge is placed at a point in an electric field, it will experience a force along the tangent of electric field line at that point and will move under its influence. The path described by this unit positive charge is called the electric field line. The direction of the force on it at that point gives the direction of the line. The direction is indicated by an arrowhead on the field line.
The tangent at any point on a field line gives the direction of the electric field intensity at that point.
Remember that the lines have no real existence. Scientist Fara-day introduced these imaginary lines to explain attraction and repulsion between two charged bodies and mentioned some definite properties of these lines.
Properties of electric field lines:
- Electric field lines start from a positive charge and end on a negative charge. There is no electric field line inside a conductor. For an isolated positive charge, the lines start from it and diverge to infinity and for an isolated negative charge, the lines of force coming from infinity and converge on the charge.
- Two field lines can never intersect. Because if they do so, then two tangents could he drawn at the point of intersection, which would indicate two directions of the electric
field intensity at a single point. But this is not possible. - Electric field lines are always normal to the surface of a charged conductor.
- The electric field lines tend to contract longitudinally. This phenomenon explains attraction between two unit charges.
- The lines tend to repel one another laterally. This explains repulsion between two like charges.
- At the two ends of each field line there should be equal and opposite charges.
- No line starts from a conductor and ends on it. From this fact it may be concluded that there are no lines inside a hollow conductor.
- A region of closely spaced field lines indicate a strong electric field; sparsely spaced lines indicate a region of weak electric field.
Maps of field lines: Maps of field lines are shown below for a few special cases.
An isolated positive charge [Fig.]: In this case, the field lines are directed away from the charge and are arranged uniformly. If they are drawn backwards, they meet at the centre of the charge.
An isolated negative charge [Fig.]: In this case, the field lines are directed towards the charge and are arranged uniformly. If the lines are extended, they meet at the centre of the
charge. Clearly the lines are similar to those in Fig., but are oppositely directed.
Two equal but opposite charges [Fig.]: Here the lines start from the positive charge and some of them end at the negative charge. Due to longitudinal contractive tendency of the lines, two opposite charges attract each other. This mapping of the lines is similar to that of a bar magnet.
Two equal and similar charges [Fig.]: Here the field lines starting from the charges repel one another and travel off to an infinite distance. At the neutral point, denoted by the × sign, the resultant intensity is zero. At this point, due to both the charges, the intensities are equal and opposite. As the lines repel laterally (i.e., sidewise), repulsion takes place between the two charges.
Uniform electric field [Fig.]: The field lines of a uniform electric field are represented by parallel equidistant straight lines. If may be noted that all the electric fields shown in the figures are non-uniform.
Electric intensity in terms of field lines: According to scientist Maxwell, the number of lines emanating from a charge q placed in a medium of permittivity ε is \(\frac{q}{\epsilon}\) (in CGS system \(\frac{4 \pi q}{k}\))
Now imagine a sphere of radius r having a charge q at its centre [Fig.] Clearly, \(\frac{q}{\epsilon}\) field lines will cross the surface area of the sphere normally. Since the surface area of the sphere is 4πr2, the number of field lines passing normally per unit area of the sphere = \(\frac{q / \epsilon}{4 \pi r^2}\) = \(\frac{q}{4 \pi \epsilon r^2}\)
But due to the charge q, the electric field at any point on the surface of the sphere = \(\frac{q}{4 \pi \epsilon r^2}\) . So we can say that the electric field at a point Is equal to the number of field lines passing normally through unit area surrounding that point.