Reading field lines

Problem

Reading field lines. The figure shows a fragment of the electric field lines generated by an unspecified charge distribution. Points $A$ and $B$ are two marked positions.

(a) At which of the two points does the electric field have the larger magnitude? Justify the criterion used. (b) Qualitatively draw the vector $\vec{E}$ at both points. (c) Indicate in which direction a negative test charge placed at $A$ would move.

Solution

(a) Where the field is more intense. The criterion is the density of the field lines: the magnitude of $\vec{E}$ is proportional to the number of lines per unit area perpendicular to them. In the figure the lines are denser (closer together) on the left, near $A$, and thin out towards the right, near $B$. Hence $\ev{|\vec{E}_A| > |\vec{E}_B|}$

(b) Direction and orientation of $\vec{E}$. At every point the vector $\vec{E}$ is tangent to the field line passing through that point and is oriented in the direction of the arrows (from left to right). At $A$ the vector is tangent to the central line, longer (intense field); at $B$ it is tangent and shorter (weak field), both directed along the lines’ direction of travel.

(c) Negative test charge at $A$. The force on a charge is $\vec{F} = q\vec{E}$. For a negative charge ($q < 0$) the force is antiparallel to the field: while the lines (and hence $\vec{E}$) point to the right, the negative charge is pushed in the direction opposite to the field lines, that is, to the left. $\ev{\vec{F} = q\vec{E}, \quad q<0 \Rightarrow \vec{F} \ \text{opposite to } \vec{E} \ (\text{to the left})}$

Links

Topics: Electric field and potential Concepts: Electric field