Question

At Los Alamos National Laboratories, one means of producing very large magnetic fields is known as the EPFCG (explosively-pumped flux compression generator), which is used to study the effects of a high-power electromagnetic pulse (EMP) in electronic warfare. Explosives are packed and detonated in the space between a solenoid and a small copper cylinder coaxial with and inside the solenoid, as shown in the figure. The explosion occurs in a very short time and collapses the cylinder rapidly. This rapid change creates inductive currents that keep the magnetic flux constant while the cylinder's radius shrinks by a factor of \(r_{\mathrm{i}} / r_{\mathrm{f}}\). Estimate the magnetic field produced, assuming that the radius is compressed by a factor of 14 and the initial magnitude of the magnetic field, \(B_{i}\), is \(1.0 \mathrm{~T}\).

Short answer

Answer: The final magnetic field produced is 196 T.

Step-by-step solution

Expression for Magnetic Flux

To begin solving this problem, we will first write the expression for magnetic flux inside a solenoid. Magnetic flux is given by: \(\Phi = nAB\) where \(n\) is the number of loops per unit length of the solenoid, \(A\) is the area of the solenoid's cross-section where the field is present, and \(B\) is the magnetic field inside the solenoid.

Use Flux Conservation

According to the problem, the magnetic flux should remain constant throughout the process. Hence, we can write: \(\Phi_i = \Phi_f\) where \(\Phi_i\) is the initial magnetic flux, and \(\Phi_f\) is the final magnetic flux.

Plug in Initial and Final Flux Expressions

Now we can write the expressions for the initial and final magnetic fluxes using the formula from step 1: \(n_i A_i B_i = n_f A_f B_f\) Since the number of loops per unit length remains unchanged, \(n_i = n_f = n\). Also, the initial magnetic field is given as \(B_i = 1.0~T\). We now have: \(nA_i (1.0~\mathrm{T}) = n A_f B_f\)

Relate Initial and Final Areas

We are given that the radius is compressed by a factor of 14 (i.e., \(\frac{r_i}{r_f} = 14\)). The initial and final areas of the solenoid's cross-section can be expressed as: \(A_i = \pi r_i^2\) and \(A_f =\pi r_f^2\) Since \(\frac{r_i}{r_f} = 14\), we can rewrite the final radius as \(r_f = \frac{r_i}{14}\). Now we can express the final area in terms of the initial area: \(A_f = \pi \left(\frac{r_i}{14}\right)^2 = \frac{1}{196}\pi r_i^2 = \frac{1}{196}A_i\)

Find the Final Magnetic Field

Now we can plug in the relationship between initial and final areas into the flux conservation equation: \(n A_i (1.0~\mathrm{T}) = n \left(\frac{1}{196}A_i\right) B_f\) We can divide both sides by \(n A_i\) and solve for \(B_f\): \(B_f = 196 (1.0~\mathrm{T})\) So, the final magnetic field produced is \(\boxed{196~\mathrm{T}}\).

Key concepts

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. In the context of a solenoid, which is a coil of wire designed to produce a magnetic field when an electric current passes through it, magnetic flux can be quantified by the equation:

• \( \Phi = nAB \)

Here, \( n \) is the number of turns (or loops) per unit length of the solenoid, \( A \) is the area that the magnetic field lines penetrate through, and \( B \) is the magnetic field itself.
The magnetic flux gives us an idea of how much magnetic field is being "threaded" through the area \( A \). The number of turns per unit length, \( n \), helps increase the magnetic field strength produced by the solenoid, thus increasing the magnetic flux.
Understanding magnetic flux is essential, as it directly relates to how effective a solenoid can be in generating a magnetic field in various applications, including in technologies like the explosively-pumped flux compression generator (EPFCG).

Solenoid

A solenoid is a type of electromagnet made by coiling a wire into a tight, cylindrical shape. When electricity is passed through the wire, it generates a magnetic field along its interior. This makes solenoids particularly useful in a variety of applications where controlled magnetic fields are required.
Solenoids come in many forms and can vary in intensity based on several parameters, such as the current flowing through them, the number of turns in the coil, and the core material placed inside the coil.
In the Los Alamos EPFCG scenario, the solenoid is used because it efficiently generates a uniform magnetic field that can be manipulated by changing the characteristics of the coil or the current passing through it. By enclosing the explosive between the solenoid and a metal cylinder, rapid changes in the solenoid environment produce significant changes in the magnetic field, which are then utilized for high-powered experiments or applications.

Flux Conservation

Flux conservation is a fundamental principle in physics stating that the magnetic flux through a closed loop remains constant unless disrupted by external fields or energies. In simpler terms, when no outside influence is acting on a magnetic loop, the amount of magnetic flux should stay the same even if the size or the orientation of the magnetic field changes.
In systems like the EPFCG at Los Alamos, this concept ensures that the intense, rapidly changing magnetic environments created by explosions do not lead to a loss of magnetic energy. The idea is that while the radius of the coil reduces (as described by the compression factor of 14), the magnetic flux remains constant.

• This principle allows engineers and scientists to predict and control the resulting magnetic field, despite dramatic changes in the solenoid's structure.

Overall, flux conservation allows for the precise engineering and predictable outcomes necessary for critical experiments.

Inductive Currents

Inductive currents are electric currents that are created in response to a changing magnetic field, as described by Faraday's Law of Induction. This phenomenon occurs because a change in the magnetic environment of a coil or loop induces an electromotive force (emf), which, in turn, generates a current.

• In practical applications, inductive currents can be utilized to create or control magnetic fields with great accuracy.
• The magnitude of the induced current depends on how quickly the magnetic field changes and the amount of turns in the coil.

In the context of the EPFCG, when the explosive detonates and compresses the copper cylinder, inductive currents are generated to oppose the change in magnetic flux, as per Lenz's Law. This means the system naturally generates currents to maintain a stable magnetic field, helping to make the explosive magnetic changes precise and usable for their intended applications. This principle of inducing currents is key for maintaining consistent magnetic fields and allows critical processes to proceed without unintended disruptions.