Figure 32-35 a shows the current i that is produced in a...

Figure $32-35 a$ shows the current $i$ that is produced in a wire of resistivity $1.62 \times 10^{-8} \Omega \cdot \mathrm{m}$. The magnitude of the current versus time $t$ is shown in Fig. $32-35 b$. The vertical axis scale is set by $i_{s}=10.0$ A, and the horizontal axis scale is set by $t_{s}=50.0 \mathrm{~ms}$. Point $P$ is at radial distance $9.00 \mathrm{~mm}$ from the wire's center. Determine the magnitude of the magnetic field $\vec{B}_{i}$ at point $P$ due to the actual current $i$ in the wire at (a) $t=20 \mathrm{~ms}$, (b) $t=$ $40 \mathrm{~ms}$, and (c) $t=60 \mathrm{~ms}$. Next, assume that the electric field driving the current is confined to the wire. Then determine the magnitude of the magnetic field $\vec{B}_{i d}$ at point $P$ due to the displacement current $i_{d}$ in the wire at (d) $t=20 \mathrm{~ms}$, (e) $t=$ $40 \mathrm{~ms}$, and (f) $t=60 \mathrm{~ms}$. At point $P$ at $t=20 \mathrm{~s}$, what is the direction (into or out of the page) of (g) $\vec{B}_{i}$ and (h) $\vec{B}_{i d}$ ?

Key Concepts

Maxwell's Correction to Ampère's Law

Maxwell's correction to Ampère’s law incorporates the displacement current term, thereby extending the law's applicability to time-dependent scenarios. This corrected version of Ampère’s law is essential for the complete formulation of electromagnetic fields, uniting the effects of both conduction and displacement currents, and is fundamental to the development of electromagnetic wave theory.

Displacement Current

Displacement current is a concept introduced by Maxwell to account for time-varying electric fields in regions where there is no physical movement of charge. It is defined as the rate of change of the electric flux and plays an integral role in ensuring the continuity of current in electromagnetic theory, especially in situations involving rapidly changing electric fields.

Right-Hand Rule

The right-hand rule is a practical mnemonic that helps determine the direction of the magnetic field induced by a current. When the thumb of the right hand points in the direction of the conventional current, the curled fingers indicate the direction of the magnetic field lines surrounding the conductor. This rule is fundamental for visualizing and predicting magnetic field directions in various electromagnetic applications.

Magnetic Field from a Current-Carrying Wire

This concept addresses how an electric current flowing through a conductor produces a magnetic field in the surrounding space. Using the relation B = ??I/(2?r) for an ideal long, straight conductor, the magnetic field (B) is directly proportional to the current (I) and inversely proportional to the distance (r) from the wire. This foundational idea stems from Ampère’s circuital law and is crucial for understanding how currents generate magnetic effects.

Ampère's Circuital Law

Ampère’s circuital law relates the integrated magnetic field around a closed loop to the total conduction current passing through the loop. This law is particularly useful in systems with high symmetry and provides a direct method to derive the magnitude of the magnetic field produced by steady currents in conductors.

Transcript

00:01 In this problem on the topic of magnetism, we are shown in the figure the current that is produced by a wire of resistivity 1 .62 times 10 to the minus 8 or meters.

00:11 The magnitude of the current versus time is shown in the next figure.

00:15 The vertical axis in that figure is set by is, which is 10 ampiers, and the horizontal axis scale is set by ts, which is 50 milliseconds.

00:24 The point p is at a radial distance 9 millimeters from the center of the wire, and we want to find the magnitude of the magnetic field b i at point p due to the actual current i in the wire firstly at 20 then 40 and then 60 milliseconds next we'll assume that the electric field driving the current is confined to the wire and we want to find the magnitude of the magnetic field b i d at point p which is due to the displacement current in the wire again at 20 40 and 60 milliseconds and then we want to find at 20 seconds the direction of b .i and bid.

01:03 Now from the figure, we can see that i is equal to 4 ampiers when the time t is 20 milliseconds.

01:18 And so from here, the magnetic field due to the actual current bi is mu -not -i over 2 pi -r, and using the information given, this is 0 .089 miltas.

01:41 Next, if we are given the time of 40 milliseconds, the current is 8 ampiers.

01:53 And so using the equation above, we get the magnetic field b .i to be 0 .18 milliseconds.

02:14 And then for part c, we have at 60 milliseconds the current i is equal to 10 amperes since the current is 10 ampiers when t is greater than 50 milliseconds which means that the displacement current or rather the magnetic field b i is equal to 0 .22 millie teslas now for part d the displacement current in terms of of the time derivative of the electric field id is equal to epsilon not a, d, d, t...

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