Electric Charges, Fields, and Gauss's Law · ⁨الشحنات الكهربائية والمجالات وقانون غاوس⁩

Learning Objective 8.1.A: Describe the electric force that results from the interactions between charged objects or systems.

• 8.1.A.1 Charge is a fundamental property of all matter.
  • 8.1.A.1.i Charge is a scalar quantity and is described as positive or negative.
  • 8.1.A.1.ii The magnitude of the charge of a single electron or proton, the elementary charge $e$, can be considered to be the smallest indivisible amount of charge.
  • 8.1.A.1.iii The charge of an electron is $-e$ and the charge of a proton is $+e$, and a neutron has no electric charge.
  • 8.1.A.1.iv A point charge is a model in which the physical size of a charged object or system is negligible in the context of the situation being analyzed.
• 8.1.A.2 Coulomb's law describes the electrostatic force between two charged objects as directly proportional to the magnitude of each of the charges and inversely proportional to the square of the distance between the objects.
  • Relevant equation: $\left| \vec{F}_E \right| = \dfrac{1}{4\pi\varepsilon_0} \dfrac{|q_1 q_2|}{r^2} = k \dfrac{|q_1 q_2|}{r^2}$
• 8.1.A.3 The direction of the electrostatic force depends on the signs of the charges of the interacting objects and is along the line of separation between the objects.
  • 8.1.A.3.i Two objects with charges of the same sign exert repulsive forces on each other.
  • 8.1.A.3.ii Two objects with charges of opposite signs exert attractive forces on each other.
• 8.1.A.4 Electric forces are responsible for some of the macroscopic properties of objects in everyday experiences. However, the large number of particle interactions that occur make it more convenient to treat everyday forces in terms of nonfundamental forces called contact forces, such as normal force, friction, and tension.

Learning Objective 8.1.B: Describe the electric and gravitational forces that result from interactions between charged objects with mass.

• 8.1.B.1 Electrostatic forces can be attractive or repulsive, while gravitational forces are always attractive.
• 8.1.B.2 For any two objects that have mass and electric charge, the magnitude of the gravitational force is usually much smaller than the magnitude of the electrostatic force.
• 8.1.B.3 Gravitational forces dominate at larger scales even though they are weaker than electrostatic forces, because systems at large scales tend to be electrically neutral.

Learning Objective 8.1.C: Describe the electric permittivity of a material or medium.

• 8.1.C.1 Electric permittivity is a measurement of the degree to which a material or medium is polarized in the presence of an electric field.
• 8.1.C.2 Electric polarization can be modeled as the induced rearrangement of electrons by an external electric field, resulting in a separation of positive and negative charges within a material or medium.
• 8.1.C.3 Free space has a constant value of electric permittivity, $\varepsilon_0$, that appears in physical relationships.
• 8.1.C.4 The permittivity of matter has a value different from that of free space that arises from the matter's composition and arrangement.
  • 8.1.C.4.i In a given material, electric permittivity is determined by the ease with which electrons can change configurations within the material.
  • 8.1.C.4.ii Conductors are made from electrically conducting materials in which charge carriers move easily; insulators are made from electrically nonconducting materials in which charge carriers cannot move easily.

Boundary statement: AP Physics C: Electricity & Magnetism only expects students to make calculations of the electric force between four or fewer interacting charged objects or systems. The analysis of the resulting electric force from more charges is allowed in situations of high symmetry. Note that students are expected to calculate the electric fields of charge distributions, as described in Topics 8.4 and 8.6.

Learning Objective 8.3.A: Describe the electric field produced by a charged object or configuration of point charges.

• 8.3.A.1 Electric fields may originate from charged objects.
• 8.3.A.2 The electric field at a given point is the ratio of the electric force exerted on a test charge at the point to the charge of the test charge.
  • Relevant equation: $\vec{E} = \dfrac{\vec{F}_E}{q}$
  • 8.3.A.2.i A test charge is a point charge of small enough magnitude such that its presence does not significantly affect an electric field in its vicinity.
  • 8.3.A.2.ii An electric field points away from isolated positive charges and toward isolated negative charges.
  • 8.3.A.2.iii The electric force exerted on a positive test charge by an electric field is in the same direction as the electric field.
• 8.3.A.3 The electric field is a vector quantity and can be represented in space using vector field maps.
  • 8.3.A.3.i The net electric field at a given location is the vector sum of individual electric fields created by nearby charged objects.
  • 8.3.A.3.ii Electric field maps use vectors to depict the magnitude and direction of the electric field at many locations within a given region.
  • 8.3.A.3.iii Electric field line diagrams are simplified models of electric field maps and can be used to determine the relative magnitude and direction of the electric field at any position in the diagram.

Learning Objective 8.3.B: Describe the electric field generated by charged conductors or insulators.

• 8.3.B.1 While in electrostatic equilibrium, the excess charge of a conductor is distributed on the surface of the conductor, and the electric field within the conductor is zero.
  • 8.3.B.1.i At the surface of a charged conductor, the electric field is perpendicular to the surface.
  • 8.3.B.1.ii The electric field outside an isolated sphere with spherically symmetric charge distribution is the same as the electric field due to a point charge with the same net charge as the sphere located at the center of the sphere.
• 8.3.B.2 While in electrostatic equilibrium, the excess charge of an insulator is distributed throughout the interior of the insulator as well as at the surface, and the electric field within the insulator may have a nonzero value.

Learning Objective 8.4.A: Describe the electric field resulting from a given charge distribution.

• 8.4.A.1 Expressions for the electric field of specified charge distributions can be found using integration and the principle of superposition.
  • Relevant equation: $\vec{E} = \dfrac{1}{4\pi\varepsilon_0} \displaystyle\int \dfrac{dq}{r^2} \hat{r}$
• 8.4.A.2 Symmetry considerations of certain charge distributions can simplify analysis of the electric field resulting from those charge distributions.

Boundary statement: AP Physics C: Electricity & Magnetism only expects students to use calculus to find the electric field resulting from the following charge distributions and locations: an infinitely long, uniformly charged wire or cylinder at a distance from its central axis, a thin ring of charge at a location along the axis of the ring, a semicircular arc or part of a semicircular arc at its center, and a finite wire or line charge at a point collinear with the line charge or at a location along its perpendicular bisector.

Learning Objective 8.5.A: Describe the electric flux through an arbitrary area or geometric shape.

• 8.5.A.1 Flux describes the amount of a given quantity that passes through a given area.
• 8.5.A.2 For an electric field $\vec{E}$ that is constant across an area $\vec{A}$, the electric flux through the area is defined as $\Phi_E = \vec{E} \bullet \vec{A}$.
  • 8.5.A.2.i The direction of the area vector is defined as perpendicular to the plane of the surface and outward from a closed surface.
  • 8.5.A.2.ii The sign of flux is given by the dot product of the electric field vector and the area vector.
• 8.5.A.3 The total electric flux passing through a surface is defined by the surface integral of the electric field over the surface.
  • Relevant equation: $\Phi_E = \displaystyle\int \vec{E} \cdot d\vec{A}$

Learning Objective 8.6.A: Describe the properties of a charge distribution by applying Gauss's law.

• 8.6.A.1 Gauss's law relates electric flux through a Gaussian surface to the charge enclosed by that surface.
  • Relevant equations: $\Phi_E = \dfrac{q_{\text{enc}}}{\varepsilon_0}$ and $\displaystyle\oint \vec{E} \cdot d\vec{A} = \dfrac{q_{\text{enc}}}{\varepsilon_0}$
• 8.6.A.2 A Gaussian surface is a three-dimensional, closed surface.
• 8.6.A.3 The total electric flux through a Gaussian surface is independent of the size of the Gaussian surface if the amount of enclosed charge remains constant.
• 8.6.A.4 Gaussian surfaces are typically constructed such that the electric field generated by the enclosed charge is either perpendicular or parallel to different regions of the Gaussian surface, resulting in a simplified surface integral.
• 8.6.A.5 If a function of charge density is given for a charge distribution, the total charge can be determined by integrating the charge density over the length (one dimension), area (two dimensions), or volume (three dimensions) of the charge distribution. For example: $Q_{\text{total}} = \displaystyle\int \rho(\vec{r})\, dV$
• 8.6.A.6 Maxwell's equations are the collection of equations that fully describe electromagnetism. Gauss's law is Maxwell's first equation.
  • Relevant equation: $\displaystyle\oint \vec{E} \cdot d\vec{A} = \dfrac{q_{\text{enc}}}{\varepsilon_0}$

Boundary statement: AP Physics C: Electricity & Magnetism only expects students to quantitatively apply Gauss's law to point charges and charge distributions that have spherical, cylindrical, or planar symmetry.