The magnetic field B at the center of a circular coil of...

The magnetic field $B$ at the center of a circular coil of wire carrying a current $I$ (as in Fig. $20-9 )$ is $$ B=\frac{\mu_{0} N I}{2 r} $$ where $N$ is the number of loops in the coil and $r$ is its radius. Suppose that an electromagnet uses a coil 1.2 $\mathrm{m}$ in diameter made from square copper wire 1.6 $\mathrm{mm}$ on a side. The power supply produces 120 $\mathrm{V}$ at a maximum power output of $4.0 \mathrm{kW} .(a)$ How many turns are needed to run the power supply at maximum power? (b) What is the magnetic field strength at the center of the coil? (c) If you use a greater number of turns and this same power supply (so the voltage remains at $120 \mathrm{V} ),$ will a greater magnetic field strength result? Explain.

The magnetic field $B$ at the center of a circular coil of wire carrying a current $I$ (as in Fig. $20-9 )$ is
$$
B=\frac{\mu_{0} N I}{2 r}
$$
where $N$ is the number of loops in the coil and $r$ is its radius. Suppose that an electromagnet uses a coil 1.2 $\mathrm{m}$ in diameter made from square copper wire 1.6 $\mathrm{mm}$ on a side. The power supply produces 120 $\mathrm{V}$ at a maximum power output of $4.0 \mathrm{kW} .(a)$ How many turns are needed to run the power supply at maximum power? (b) What is the magnetic field strength at the center of the coil? (c) If you use a greater number of turns and this same power supply (so the voltage remains at $120 \mathrm{V} ),$ will a greater magnetic field strength result? Explain.

Key Concepts

Trade-offs in Electromagnet Design

When designing an electromagnet, there is an inherent trade-off: increasing the number of turns boosts the magnetic field if the current is kept constant, but it also increases the resistance of the coil, thereby reducing the current available from a fixed voltage source. This balance between magnetic field enhancement and current reduction is critical when attempting to maximize the magnetic field strength, particularly under power-constrained conditions.

Resistance of a Wire and Geometrical Considerations

The resistance of a wire is determined by its material properties and dimensions, according to R = ?L/A, where ? is the resistivity of the material, L is the length of the conductor, and A is its cross-sectional area. In the design of electromagnets, incorporating more turns increases the total length of wire used, which in turn increases the resistance. This interplay between the number of turns and the resulting resistance has a significant effect on the performance of the electromagnet.

Ohm's Law and the Power Relationship

Ohm's law (V = IR) underpins the relationship between voltage, current, and resistance in an electrical circuit, while the power relationship (P = VI or P = V²/R) connects the electrical power to these quantities. In circuits powered by a voltage source, increases in resistance will reduce current for a fixed voltage, which in turn affects power delivery. Understanding these relationships is crucial when designing circuits to operate within the limits of a given power supply.

Magnetic Field of a Circular Coil

This concept deals with the magnetic field produced at the center of a circular loop or coil. The formula B = (??N I) / (2r) shows that the magnetic field is directly proportional to both the number of loops (N) and the current (I) through them, while it is inversely proportional to the radius (r) of the loop. The derivation of this formula relies on the Biot–Savart law, which describes how currents create magnetic fields through integration over the loop geometry.

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