Biot-Savart Law (Calculus): Videos & Practice Problems

The Biot-Savart law describes the magnetic field generated by a current-carrying wire. The magnetic field at a position r is given by μ4π ∫ I⁢𝛩r. This integral accounts for the direction of the current and the position vector. For a moving point charge, the law simplifies to μ4π ∫ qv𝛩r, illustrating the relationship between current and magnetic fields.

Biot-Savart Law with Calculus

The Biot-Savart law is a fundamental principle in electromagnetism that describes the magnetic field generated by a current-carrying wire. When analyzing the magnetic field at a point located a distance represented by the vector \(\mathbf{r}\) from a wire carrying a current \(I\), the law is expressed mathematically as:

\[ \mathbf{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3} \]

In this equation, \(\mu_0\) is the permeability of free space, \(d\mathbf{l}\) is an infinitesimally small vector in the direction of the current, and \(\mathbf{r}\) is the position vector from the wire to the point where the magnetic field is being measured. The cross product \(d\mathbf{l} \times \mathbf{r}\) indicates that the direction of the magnetic field is perpendicular to both the current direction and the line connecting the wire to the point of interest.

To illustrate the application of the Biot-Savart law, consider the scenario of a single moving charge \(q\). The current \(I\) can be defined as the rate of charge flow, given by \(I = \frac{dq}{dt}\). By substituting this definition into the Biot-Savart law, we can derive the magnetic field produced by a moving point charge:

\[ \mathbf{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3} = \frac{\mu_0}{4\pi} \int \frac{dq/dt \, d\mathbf{l} \times \mathbf{r}}{r^3} \]

Using implicit differentiation, we can rearrange the terms to express the magnetic field in terms of the charge's velocity \(\mathbf{v}\) and the sine of the angle \(\theta\) between the velocity vector and the position vector:

\[ \mathbf{B} = \frac{\mu_0}{4\pi} \frac{q \mathbf{v} \times \mathbf{r}}{r^2} \]

This equation simplifies to:

\[ \mathbf{B} = \frac{\mu_0}{4\pi} \frac{q v \sin \theta}{r^2} \]

Here, \(v\) is the speed of the charge, and \(\theta\) is the angle between the velocity vector and the line connecting the charge to the point where the magnetic field is measured. This derivation shows how the Biot-Savart law can be applied to both continuous current distributions and discrete moving charges, highlighting its versatility in calculating magnetic fields in various scenarios.

Magnetic Field due to Finite Wire

To determine the magnetic field generated by a finite current-carrying wire, we can utilize the Biot-Savart Law, which states that the magnetic field contribution from a small segment of current is given by:

dB = \frac{\mu_0}{4\pi} \frac{I \, d\ell \times \mathbf{r}}{r^3}

In this scenario, we consider a wire of length \(2a\) centered along the z-axis, extending from \(-a\) to \(+a\). Each segment of the wire contributes a small magnetic field, and we denote a segment in the x-direction as \(dx\). The distance \(r\) from the segment to the point of interest is defined as:

r = \sqrt{x^2 + z^2}

To simplify the expression for the magnetic field, we can express the sine of the angle \(\theta\) in terms of the opposite side (vertical distance \(z\)) and the hypotenuse \(r\):

\sin(\theta) = \frac{z}{r} = \frac{z}{\sqrt{x^2 + z^2}}

Substituting this into the expression for \(dB\), we have:

dB = \frac{\mu_0}{4\pi} \frac{I \, z}{(x^2 + z^2)^{3/2}} \, dx

Next, we integrate this expression from \(-a\) to \(+a\) to find the total magnetic field \(B\) at the point located a distance \(z\) from the wire:

B = \int_{-a}^{a} \frac{\mu_0 I z}{4\pi (x^2 + z^2)^{3/2}} \, dx

This integral can be evaluated using trigonometric substitution, leading to the result:

B = \frac{\mu_0 I}{4\pi} \cdot \frac{2a}{z \sqrt{z^2 + a^2}}

To analyze the behavior of this result as the length of the wire approaches infinity, we consider the limit as \(a\) approaches infinity. The expression simplifies to:

B = \frac{\mu_0 I}{2\pi z}

This final expression corresponds to the magnetic field produced by an infinitely long straight wire carrying current \(I\), confirming that the finite wire's magnetic field converges to the known result for an infinite wire as its length increases.

What is the magnetic field at the center of the following ring of current?

• A. μ₀i/(2r)
• B. μ₀i/(πr²)
• C. μ₀i/(2πr)
• D. μ₀i / √(2πr)

Answer: (A)

Here’s what students ask on this topic:

What is the Biot-Savart law in calculus form?

The Biot-Savart law in its calculus form describes the magnetic field B generated by a current-carrying wire. It is given by:

μo4π∫I𝛩r/r3

where μo is the permeability of free space, I is the current, dl is an infinitesimal vector element of the wire in the direction of the current, and r is the position vector from the wire element to the point where the magnetic field is being calculated. The cross product 𝛩 accounts for the direction of the magnetic field.

How does the Biot-Savart law apply to a moving point charge?

For a moving point charge, the Biot-Savart law simplifies to:

μo4π∫qv𝛩r/r3

where q is the charge, v is the velocity of the charge, and r is the position vector from the charge to the point where the magnetic field is being calculated. This form shows the relationship between the velocity of the charge and the magnetic field it generates, with the cross product 𝛩 indicating the direction of the magnetic field.

What is the significance of the cross product in the Biot-Savart law?

The cross product in the Biot-Savart law, denoted as 𝛩, is significant because it determines the direction of the magnetic field generated by the current. Specifically, the cross product dl𝛩r involves the infinitesimal current element dl and the position vector r. The result of this cross product is a vector that is perpendicular to both dl and r, indicating the direction of the magnetic field according to the right-hand rule. This directional information is crucial for accurately determining the magnetic field's orientation in space.

How do you derive the Biot-Savart law for a point charge from its general form?

To derive the Biot-Savart law for a point charge from its general form, start with the general equation:

μo4π∫I𝛩r/r3

Substitute the definition of current I=dqdt into the equation:

μo4π∫dqdt𝛩r/r3

Rearrange the differentials using implicit differentiation:

μo4π∫dq𝛩dl/r3

Recognize that dl is the distance in the direction of the charge's velocity v:

μo4π∫dqv𝛩r/r3

Integrate over the charge q to get:

μo4πqv𝛩r/r2

This is the Biot-Savart law for a moving point charge.

What are some common applications of the Biot-Savart law?

The Biot-Savart law is widely used in various applications to calculate magnetic fields. Some common applications include:

• Magnetic Field of a Current-Carrying Wire: It helps determine the magnetic field around straight or curved current-carrying conductors.
• Magnetic Field of a Circular Loop: It is used to calculate the magnetic field at the center and along the axis of a current-carrying loop.
• Electromagnetic Devices: The law is essential in designing and analyzing the magnetic fields in devices like solenoids, electromagnets, and inductors.
• Particle Physics: It helps in understanding the magnetic fields generated by moving charged particles, which is crucial in particle accelerators and detectors.

These applications highlight the importance of the Biot-Savart law in both theoretical and practical aspects of electromagnetism.