Electromagnetism and Wave Motion Final Exam Review

What is a vector quantity?

A quantity with both magnitude and direction.

What is a scalar quantity?

A quantity with magnitude only and no direction.

What are the Cartesian unit vectors?

\mathbf{i}, \mathbf{j}, \mathbf{k}

What direction does \mathbf{i} point?

The positive x-direction.

What direction does \mathbf{j} point?

The positive y-direction.

What direction does \mathbf{k} point?

The positive z-direction.

What is a unit vector?

A vector with magnitude 1 used to indicate direction.

What is the radial unit vector?

\hat{r}, a unit vector pointing radially outward from the center.

How do you add vectors in component form?

Add corresponding x, y, and z components separately.

What is the magnitude of vector \vec{r} = r_x\mathbf{i} + r_y\mathbf{j} + r_z\mathbf{k}?

|\vec{r}|=\sqrt{r_x^2+r_y^2+r_z^2}

What happens when a vector is multiplied by a positive scalar?

Its magnitude changes and its direction stays the same.

What happens when a vector is multiplied by a negative scalar?

Its direction reverses.

What is the dot product of two vectors?

A scalar equal to \vec{a}\cdot\vec{b}=ab\cos\phi.

What does the dot product physically represent?

The component of one vector in the direction of the other times the other magnitude.

Is the dot product commutative?

Yes, \vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}.

How do you calculate the dot product from components?

\vec{a}\cdot\vec{b}=a_xb_x+a_yb_y+a_zb_z

What is \mathbf{i}\cdot\mathbf{i}?

1

What is \mathbf{j}\cdot\mathbf{j}?

1

What is \mathbf{k}\cdot\mathbf{k}?

1

What is \mathbf{i}\cdot\mathbf{j}?

0

What is \mathbf{j}\cdot\mathbf{k}?

0

What is \mathbf{i}\cdot\mathbf{k}?

0

How can you find the angle between two vectors using the dot product?

\cos\phi=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}

What is the cross product of two vectors?

A vector equal to \vec{a}\times\vec{b} with magnitude ab\sin\phi and direction from the right-hand rule.

Is the cross product commutative?

No, \vec{a}\times\vec{b}=-(\vec{b}\times\vec{a}).

What is the direction of the cross product?

Perpendicular to the plane containing the two vectors.

How do you determine the direction of a cross product?

Use the right-hand rule: fingers along the first vector, curl toward the second, thumb gives the direction.

What is the magnitude of the cross product?

|\vec{a}\times\vec{b}|=ab\sin\phi

When is the magnitude of a cross product maximum?

When the angle between the vectors is 90^\circ.

What is \mathbf{i}\times\mathbf{j}?

\mathbf{k}

What is \mathbf{j}\times\mathbf{k}?

\mathbf{i}

What is \mathbf{k}\times\mathbf{i}?

\mathbf{j}

What is \mathbf{j}\times\mathbf{i}?

-\mathbf{k}

What is \mathbf{k}\times\mathbf{j}?

-\mathbf{i}

What is \mathbf{i}\times\mathbf{k}?

-\mathbf{j}

What is \vec{a}\times\vec{a}?

0

What is torque in vector form?

\vec{\tau}=\vec{r}\times\vec{F}

What is the magnitude of torque?

|\tau|=rF\sin\theta

When is torque maximum?

When the force is perpendicular to the position vector.

Why is torque a vector?

Because it has both magnitude and direction along the axis of rotation.

What are the two types of electric charge?

Positive and negative.

What is the SI unit of charge?

The coulomb (\mathrm{C}).

What is the elementary charge?

e=1.602\times10^{-19}\ \mathrm{C}

What is the charge of a proton?

+1.602\times10^{-19}\ \mathrm{C}

What is the charge of an electron?

-1.602\times10^{-19}\ \mathrm{C}

What does it mean that charge is quantized?

Any free charge is an integer multiple of the elementary charge.

What does it mean that charge is conserved?

The net charge of an isolated system remains constant.

What is an anion?

A negatively charged ion with extra electrons.

What is a cation?

A positively charged ion with missing electrons.

How do you find the number of electrons corresponding to a charge Q?

N_e=\dfrac{|Q|}{e}

What is Coulomb’s law in magnitude form?

F=k\dfrac{|q_1q_2|}{r^2}

What is the value of Coulomb’s constant?

k=8.99\times10^9\ \mathrm{N\cdot m^2/C^2}

How does the Coulomb force depend on distance?

It is inversely proportional to the square of the distance.

What happens between like charges?

They repel.

What happens between opposite charges?

They attract.

What principle is used when several charges act on one charge?

The principle of superposition.

What is an electric field?

A field that gives the force per unit positive test charge at a point.

What is the definition of electric field?

\vec{E}=\dfrac{\vec{F}}{q}

What are the SI units of electric field?

\mathrm{N/C} or \mathrm{V/m}

What is the electric field of a point charge?

\vec{E}=k\dfrac{q}{r^2}\hat{r}

What direction does the electric field point for a positive point charge?

Radially outward.

What direction does the electric field point for a negative point charge?

Radially inward.

What is an electric field line?

A line drawn so the tangent gives the direction of the electric field.

Where do electric field lines start and end?

They start on positive charges and end on negative charges.

What does a denser pattern of electric field lines mean?

A stronger electric field.

What is an electric dipole?

Two equal and opposite charges separated by a distance.

What is the electric dipole moment?

\vec{p}=q\vec{d}, directed from negative to positive charge.

What torque acts on an electric dipole in an electric field?

\vec{\tau}=\vec{p}\times\vec{E}

What is the electric potential energy of a dipole in an electric field?

U=-\vec{p}\cdot\vec{E}

What happens to a dipole in a uniform electric field?

It tends to rotate to align with the field.

Does a uniform electric field produce a net force on an ideal dipole?

No, only a torque.

What is linear charge density?

\lambda=\dfrac{Q}{L}

What are the units of linear charge density?

\mathrm{C/m}

What is surface charge density?

\sigma=\dfrac{Q}{A}

What are the units of surface charge density?

\mathrm{C/m^2}

What is volume charge density?

\rho=\dfrac{Q}{V}

What are the units of volume charge density?

\mathrm{C/m^3}

What is dq for a linear charge distribution?

dq=\lambda\,ds

What is dq for a surface charge distribution?

dq=\sigma\,dA

What is dq for a volume charge distribution?

dq=\rho\,dV

Why do continuous charge distributions require integration?

Because the charge is spread over a line, area, or volume, so many small contributions must be summed.

What is electric flux?

The amount of electric field passing through a surface.

What is the expression for electric flux?

\Phi_E=\int \vec{E}\cdot d\vec{A}

What does the dot product in electric flux mean?

Only the component of \vec{E} perpendicular to the surface contributes.

What is Gauss’s law?

The total electric flux through a closed surface equals q_{\text{enc}}/\varepsilon_0.

Write Gauss’s law.

\oint \vec{E}\cdot d\vec{A}=\dfrac{q_{\text{enc}}}{\varepsilon_0}

What is \varepsilon_0?

The permittivity of free space, 8.85\times10^{-12}\ \mathrm{C^2/(N\cdot m^2)}.

When is Gauss’s law especially useful?

When the charge distribution has high symmetry such as spherical, cylindrical, or planar symmetry.

What Gaussian surface is best for a long charged wire?

A cylinder coaxial with the wire.

What electric field is produced by an infinite line of charge?

E=\dfrac{\lambda}{2\pi\varepsilon_0 r}

What Gaussian surface is best for an infinite sheet of charge?

A pillbox straddling the sheet.

What is the field of a non-conducting infinite sheet of charge?

E=\dfrac{\sigma}{2\varepsilon_0}

What is the field of a conducting infinite sheet of charge?

E=\dfrac{\sigma}{\varepsilon_0}

What is electric potential?

Electric potential energy per unit charge.

What is the definition of electric potential?

V=\dfrac{U}{q}

What is the SI unit of electric potential?

The volt (\mathrm{V}), equal to \mathrm{J/C}.

What is the electric potential of a point charge?

V=k\dfrac{q}{r}

Is electric potential a scalar or vector?

Scalar.

How are electric field and electric potential related?

The field points in the direction of decreasing potential.

What is the relation between potential difference and field?

V_f-V_i=-\int_i^f \vec{E}\cdot d\vec{s}