Pho.rs: Fields of charged bodies

Условие

Part A. Ring

Part B. Two halves of ring

Part C. Cyclinder

Русский

Part A. Ring (2.0 points)

A ring of radius $R$ is uniformly charged with a linear charge density $\lambda$.

A1 Determine the magnitude of the electric field at the center of the charged ring.

A2 Determine the electric potential on the axis of the ring at a distance $x$ from its center.

A3 Determine the magnitude of the electric field on the axis of the ring at a distance $x$ from its center.

A4 For which value of $x$ the magnitude of the electric field reaches its maximum? Also, determine the magnitude of the electric field in this point.

A5 Determine the magnitude of the electric field created by the disk on the axis at a distance of $x$ from its center. Disk is uniformly charged with the surface density of $\sigma$ and has a radius $R$.

Part B. Two halves of ring (2.0 points)

A ring is created of two halves of radius $R$. Both are uniformly charged but while one half has a linear charge density of $\lambda$, the other has $-\lambda$.

B1 Determine the electric potential on the axis of the ring at a distance $x$ from its center.

B2 Determine the magnitude of the electric field in the center of the ring.

B3 Determine the magnitude of the electric field created by the disk on the axis at a distance of $x$ from its center. Disk is uniformly charged with the surface density of $\sigma$ and has a radius $R$.

Part C. Cyclinder (6.0 points)

An infinitely long thin-walled cylinder of radius $R$ uniformly charged with a surface charge density $\sigma$.

C1 What is the magnitude of the electric field at the distance $r<R$ from the axis of the cylinder?

Let us consider a semi-infinite thin-walled cylinder of radius $R$, which is uniformly charged with a surface charge density $\sigma$. In the next questions you will determine the electric field at each point on its base (see fig.).

C2 Determine the magnitude of the electric field in the center $O$ of the cylinder's base.

Найдите модуль напряженности поля в центре основания цилиндра $O$.

C3 Consider a point $A$ on the cylinder's base which is located at the distance $r<R$ from $O$. Determine the projection of the electric field on the line $OA$.

To answer the next question it may be useful to know the dependence of the radial electric field created by the ring in its plane. Consider a ring of radius $R$ and charge $Q$. At a distance $r$ from its center the radial electric field is given by
$$E_r=\frac{kQ}{2{\pi}R^2}\cdot y(x),$$where $x=r/R$. The graph of $y$ vs $x$ is represented below.

C4 Determine the magnitude of the electric field created by the semi-inifinite cylinder at the point $A$ which is located at a distance of $r=0.9R$ from point $O$.