AP Physics C: Electricity and Magnetism Practice Quiz: Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law

Correct Answer: B

The provided content explicitly states that the Biot-Savart law defines the magnitude and direction of a magnetic field created by an electrical current.

Correct Answer: C

The force is given by $\vec{F}_{B}=I\vec{l}\times\vec{B}$. The magnitude of the cross product is $|\vec{l}| |\vec{B}| \sin(\theta)$, which is maximum when $\sin(\theta)=1$. This occurs when the angle $\theta$ is 90° (the wire is perpendicular to the field).

Correct Answer: B

The cross product of two vectors results in a third vector that is perpendicular to the plane containing the original two. In this equation, the cross product $d\vec{l}\times\hat{r}$ determines the direction of the infinitesimal magnetic field vector $d\vec{B}$.

Correct Answer: B

The force is given by $\vec{F}_{B}=I\vec{l}\times\vec{B}$. The magnitude of the cross product is zero when the vectors $\vec{l}$ and $\vec{B}$ are parallel (angle of 0°) or anti-parallel (angle of 180°). In either case, the wire is parallel to the field.

Correct Answer: C

The Biot-Savart law, $d\vec{B}=\frac{\mu_{0}}{4\pi}\frac{I(d\vec{l}\times\hat{r})}{r^{2}}$, involves the cross product $d\vec{l}\times\hat{r}$. A property of the cross product is that the resulting vector ($d\vec{B}$ in this case) is perpendicular to both of the original vectors ($d\vec{l}$ and $\hat{r}$).

Correct Answer: C

The Biot-Savart law, $d\vec{B}=\frac{\mu_{0}}{4\pi}\frac{I(d\vec{l}\times\hat{r})}{r^{2}}$, shows that the magnitude of the magnetic field contribution is inversely proportional to the square of the distance ($r^2$). If $r$ is doubled, $r^2$ becomes $(2r)^2 = 4r^2$, so the magnitude of $d\vec{B}$ is divided by 4 (quartered).

Correct Answer: B

Using the right-hand rule for the force equation $\vec{F}_{B}=I\vec{l}\times\vec{B}$: point your fingers in the direction of the current (east), curl them towards the direction of the magnetic field (north). Your thumb will point up, indicating the direction of the force.

Correct Answer: B

For a circular loop, every point on the wire ($d\vec{l}$) is equidistant from the center. This constant distance, $r$, is the radius of the loop. This simplifies the integration of the Biot-Savart law because $r$ can be treated as a constant and taken out of the integral. Additionally, the angle between $d\vec{l}$ and $\hat{r}$ is always 90 degrees.

Correct Answer: C

The content explicitly states that this equation describes the force exerted on current-carrying wires by a magnetic field.

Correct Answer: A

The Biot-Savart law is used to calculate the magnetic field ($d\vec{B}$) *created by* a current ($I d\vec{l}$). The force law is used to calculate the force ($\vec{F}_{B}$) *exerted on* a current ($I\vec{l}$) that is placed in a pre-existing external magnetic field ($\vec{B}$).

Correct Answer: B

The first point of the provided content states that the topic is to 'Describe the magnetic field produced by a current-carrying wire.'

Correct Answer: B

The magnitude of the cross product $d\vec{l}\times\hat{r}$ is $|d\vec{l}| |\hat{r}| \sin(\theta)$, where $\theta$ is the angle between the vectors. If the angle is 0°, then $\sin(0°)=0$. This means the cross product is zero, and therefore the magnetic field contribution $d\vec{B}$ is zero. This occurs for points directly in line with the current element.

Correct Answer: C

The Biot-Savart law, $d\vec{B}=\frac{\mu_{0}}{4\pi}\frac{I(d\vec{l}\times\hat{r})}{r^{2}}$, shows that the magnetic field vector $d\vec{B}$ is directly proportional to the current element vector $I d\vec{l}$. Reversing the current $I$ is equivalent to reversing the direction of the vector $d\vec{l}$. This reverses the direction of the cross product, and thus reverses the direction of the magnetic field.

Correct Answer: D

The Biot-Savart law gives the magnetic field contribution, $d\vec{B}$, from an infinitesimal segment of wire, $d\vec{l}$. To find the total magnetic field $\vec{B}$ from a wire of finite length, one must perform a vector sum (an integral) of all the infinitesimal contributions along the entire length of the wire. This is a direct application of the principle of superposition.