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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 22.4 Electric Field of Many Charges

The electric field is a vector field and vectors obey the principle of superposition. If multiple point charges exist in some region of space, and we want to know the electric field at some point in space in the vicinity of these charges, then we need to sum the vector components of the electric field from each charge at that point in space to determine the total electric field. It may be useful for you to review vector addition before moving forward.
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Definition 22.4.1. Electric Field of Many Point Charges.

The electric field due to multiple point charges is the vector sum
\begin{equation*} \vec{E}_{tot} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + ... = \Sigma{\vec{E}_i} \text{.} \end{equation*}
where \(i \) refers to the electric field from the ith charge in the sum.
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q1, 12, and q3, each with their own r-hat vector pointing toward point P, corresponding to three electric fields at point P.
Figure 22.4.2. A visualization of the electric field at a point in space due to multiple charges.
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Subsubsection Activities

Activity 22.4.1. Two Protons.

Consider two protons separated by a distance \(d \) on the \(x\)-axis equidistant from the origin.
  1. Overlaid on the appropriate coordinate system, sketch a diagram of the physical situation and label all quantities of interest.
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  2. Considering the principle of superposition, if you are at a point on the positive \(y\)-axis, in which direction do you expect the net electric field to point? Back up your argument with a diagram.
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  3. Use your coordinate system and defined variables from your diagram to construct a symbolic expression for the electric field \(\vec{E}(y)\) points along the \(y\)-axis of your coordinate system.
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  4. What do you expect the field to look like if you move very far up the \(y\)-axis such that \(y \gg d\text{?}\)
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