Relation Cross-Scans

Field Potential Superposition Cross-Scan

arrows and scalars

At one point, fields add as signed arrows while potentials add as signed scalars. Side-by-side rows make the difference scanable.

electrostatics Compare the field ledger and the potential ledger at the same point.

Example

At one point, fields add as signed arrows while potentials add as signed scalars. Side-by-side rows make the difference scanable. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Use one point for both ledgers

At the same point P, field is a signed arrow ledger and potential is a signed scalar ledger.

same point P:Enet and Vnet\text{same point }P:\quad E_{\text{net}}\text{ and }V_{\text{net}}
Field and potential cross-scanEqual field arrows cancel at the sample point.P+4 Cleft+3 Cright4 N/C4 N/C

Add field arrows with signs

The middle row has a rightward contribution of 4 newtons per coulomb and a leftward contribution written as -4 newtons per coulomb. The net field is zero.

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 cancellationNo net field arrow remains in the middle row.P+4 Cleft+3 Cright4 N/C4 N/C

Add potential without arrows

For potential, the same middle row adds positive scalar contributions. The net potential is 12 volts even when field cancels.

Vnet=12 VV_{\text{net}}=12\ \text{V}

Scan both ledgers side by side

Three rows make the distinction visible. The field column can cross through zero while the potential column keeps increasing.

ELEREnetVLVRVnet4 N/C−2 N/C2 N/C6 V3 V9 V4 N/C−4 N/C0 N/C6 V6 V12 V4 N/C−6 N/C−2 N/C6 V9 V15 V\begin{array}{c|c|c|c|c|c}E_L&E_R&E_{\text{net}}&V_L&V_R&V_{\text{net}}\\4\ \text{N/C}&-2\ \text{N/C}&2\ \text{N/C}&6\ \text{V}&3\ \text{V}&9\ \text{V}\\4\ \text{N/C}&-4\ \text{N/C}&0\ \text{N/C}&6\ \text{V}&6\ \text{V}&12\ \text{V}\\4\ \text{N/C}&-6\ \text{N/C}&-2\ \text{N/C}&6\ \text{V}&9\ \text{V}&15\ \text{V}\\\end{array}

Choose the ledger that answers the question

Use field for force direction on a positive test charge. Use potential for energy per coulomb. The same sources can give different-looking rows because arrows and scalars are different objects.

force direction⇒Eenergy per charge⇒V\text{force direction}\Rightarrow E\qquad \text{energy per charge}\Rightarrow V
Field and potential cross-scanThe diagram is the zero-field middle row.P+4 Cleft+3 Cright4 N/C4 N/C