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Electric field and zero electric field regions

Electric field is a vector quantity that describes the force per unit charge experienced by a positive test charge placed in a given point in space. The electric field can be calculated from the electric potential, which depends on the charge distribution that creates the field. In some cases, the electric field can be zero in certain regions, due to symmetry, cancellation, or other factors. In this answer, we will explore some examples of arrangements that can produce zero electric field regions.

Uniformly charged sphere

A uniformly charged sphere has a spherically symmetric charge distribution, which means that the electric field it creates is also spherically symmetric. However, at the center of the sphere, the electric field is zero, because any charge element on the sphere that contributes to the field at that point is canceled by an opposite charge element on the opposite side of the sphere. Therefore, the electric field is zero at the center of a uniformly charged sphere.

Electric dipole

An electric dipole is a pair of equal and opposite charges separated by a small distance. The electric field it creates depends on the distance from the dipole axis and the orientation of the dipole. Along the dipole axis, the electric field is zero at the midpoint between the charges, because the contributions from the two charges cancel each other out. This is a case of symmetry, where the electric field is zero due to the arrangement of charges.

Charged conducting shell

A charged conducting shell has a uniform charge distribution on its surface, due to the repulsion of like charges. Inside the shell, the electric field is zero, because any charge element inside the shell is surrounded by an equal amount of charge, which cancels out the electric field. This is a case of electrostatic shielding, where a conductor can protect the inside from external electric fields.

Charged conducting sphere

A charged conducting sphere has a uniform charge distribution on its surface, similar to a charged conducting shell. However, unlike the shell, the sphere can have non-zero electric field inside, if there is a charge or a cavity inside the sphere. If the sphere is hollow, the electric field is zero everywhere inside the cavity, because there is no charge inside. If the sphere has a charge inside, the electric field can be zero at the center of the sphere, if the charge is at the center or if the charge distribution is spherically symmetric.

Non-uniformly charged disk

A non-uniformly charged disk has a charge distribution that depends on the distance from the center. If the disk has a radial symmetry, such as a circular disk, the electric field can be zero at the center of the disk, if the charge distribution is chosen appropriately. This requires that the electric field contributions from the charges on the opposite sides of the disk cancel each other out, which depends on the charge density profile.

Conclusion

Electric field can be zero in certain regions due to symmetry, cancellation, or other factors. Some examples of arrangements that can produce zero electric field regions are a uniformly charged sphere at the center, an electric dipole on the axis, a charged conducting shell or sphere inside, and a non-uniformly charged disk at the center. Understanding these examples can help us appreciate the complexity and beauty of electrostatics, and apply them to practical problems.