State Gauss's law in terms of D for a dielectric.

State Gauss's law in terms of D for a dielectric.

• A. ∮ D · dA = Q_free_enclosed.
• B. ∮ D · dA = Q_bound_enclosed.
• C. ∮ E · dA = Q_free_enclosed/ε0.
• D. ∮ D · dA = Q_total_enclosed.

Answer: (A)

In dielectrics, the displacement field D is defined to capture the material’s response (polarization) so that Gauss’s law can separate free charges from bound charges. The law in terms of D says that the net flux of D through any closed surface equals the free charge enclosed by that surface: ∮ D · dA = Q_free_enclosed. Polarization in the dielectric creates bound charges, which come from the material itself, not from external free charges. The way D is defined ensures those bound charges don’t add to the Gauss flux you compute with D; only the free charges contribute to ∮ D · dA. That’s why this form is so useful: it lets you work with the free charges you actually place on conductors without getting tangled in the bound charges produced by the dielectric’s polarization. To contrast the other ideas briefly: using D to equal the bound charge enclosed would ignore the free charges that are still influencing the field inside the dielectric; using E with the enclosed free charge divided by ε0 would miss the total effect of all charges (free plus bound) on the field; and equating the D flux to the total enclosed charge would ignore how polarization reshapes the field so that D only counts free charges. The correct relation, ∮ D · dA = Q_free_enclosed, cleanly separates the free sources from the bound charges in a dielectric.

In dielectrics, the displacement field D is defined to capture the material’s response (polarization) so that Gauss’s law can separate free charges from bound charges. The law in terms of D says that the net flux of D through any closed surface equals the free charge enclosed by that surface: ∮ D · dA = Q_free_enclosed.

Polarization in the dielectric creates bound charges, which come from the material itself, not from external free charges. The way D is defined ensures those bound charges don’t add to the Gauss flux you compute with D; only the free charges contribute to ∮ D · dA. That’s why this form is so useful: it lets you work with the free charges you actually place on conductors without getting tangled in the bound charges produced by the dielectric’s polarization.

To contrast the other ideas briefly: using D to equal the bound charge enclosed would ignore the free charges that are still influencing the field inside the dielectric; using E with the enclosed free charge divided by ε0 would miss the total effect of all charges (free plus bound) on the field; and equating the D flux to the total enclosed charge would ignore how polarization reshapes the field so that D only counts free charges. The correct relation, ∮ D · dA = Q_free_enclosed, cleanly separates the free sources from the bound charges in a dielectric.