Superposition Audits

Field Cancels, Potential Adds

same point, different ledgers

Equal sources can make the net field zero at a point while the potential remains nonzero, because vectors and scalars add differently.

Example

Equal sources can make the net field zero at a point while the potential remains nonzero, because vectors and scalars add differently. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Place equal positive sources around P

Put equal positive sources on opposite sides of point P. Their field arrows at P point in opposite directions, but their potential contributions are both positive.

same point P: compare field vectors and potential scalars\text{same point }P\text{: compare field vectors and potential scalars}
Equal positive sourcesEqual positive sources bracket the sample point.P+5 Cleft+5 Cright4 N/C4 N/C

The field vectors cancel

The left source contributes 4 newtons per coulomb to the right. The right source contributes 4 newtons per coulomb to the left, written as -4 newtons per coulomb.

Enet=4 N/C+−4 N/C=0 N/CE_{\text{net}} = 4\ \text{N/C} + -4\ \text{N/C} = 0\ \text{N/C}
Field cancelsNo net field arrow is drawn because the vector sum is zero.P+5 Cleft+5 Cright4 N/C4 N/C

The potentials still add

The potential contributions do not carry left or right arrows. The left source contributes 5 volts and the right source contributes 5 volts.

Vnet=5 V+5 V=10 VV_{\text{net}} = 5\ \text{V} + 5\ \text{V} = 10\ \text{V}
Potential addsPotential has no direction arrow, so the positive contributions add.P+5 Cleft+5 Cright4 N/C4 N/C

Zero field does not mean zero potential

At this point, the net field is 0 newtons per coulomb, but the net potential is 10 volts. These are different physical claims.

quantityoperationnet at PEsigned vector sum0 N/CVsigned scalar sum10 V\begin{array}{c|c|c}\text{quantity}&\text{operation}&\text{net at }P\\E&\text{signed vector sum}&0\ \text{N/C}\\V&\text{signed scalar sum}&10\ \text{V}\end{array}
Compare field and potentialThe same sources give zero net field but nonzero potential.P+5 Cleft+5 Cright4 N/C4 N/C

Use the right ledger for the question

Use the field ledger when asking which way a positive test charge is pushed. Use the potential ledger when asking how much energy per coulomb exists at the point.

force question⇒Eenergy-per-charge question⇒V\text{force question} \Rightarrow E\qquad \text{energy-per-charge question} \Rightarrow V
Compare field and potentialSame point, same sources, two different ledgers.P+5 Cleft+5 Cright4 N/C4 N/C
electrostatics Use the same sources to compare a vector field ledger with a scalar potential ledger.