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.

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