A cast steel core with a uniform cross-sectional area of 2 cm2 and a mean length of 25 cm is shown in Fig. The relative permeability of the steel is 48000. The air gap is 1 mm wide... A cast steel core with a uniform cross-sectional area of 2 cm2 and a mean length of 25 cm is shown in Fig. The relative permeability of the steel is 48000. The air gap is 1 mm wide and the coil has 5000 turns. Determine the current in the coil to produce a flux density of 0.80 T in the air gap.
Understand the Problem
The question is asking us to determine the current needed in a coil to achieve a specific magnetic flux density in an air gap, given parameters such as the core's properties and configuration. We will use the formula for magnetomotive force and the relationship between current, turns of the coil, and flux density to solve for the current.
Answer
The current required can be calculated using the formula $I = \frac{B \cdot l}{N \cdot \mu_0}$.
Answer for screen readers
The value of the current needed in the coil is given by:
$$ I = \frac{B \cdot l}{N \cdot \mu_0} $$
Substituting the actual values will yield the numerical answer for $I$.
Steps to Solve
- Identify Given Values
List the known values provided in the problem, such as the number of turns of the coil ($N$), the length of the coil ($l$), the desired magnetic flux density ($B$), and the permeability of free space ($\mu_0$).
- Use the Magnetic Flux Density Equation
The relationship between magnetic flux density, magnetomotive force (MMF), and permeability can be expressed as: $$ B = \mu \frac{MMF}{l} $$
Here, $MMF = N \cdot I$ where $I$ is the current. Therefore, we can rewrite the equation as: $$ B = \mu_0 \frac{N \cdot I}{l} $$
- Solve for Current (I)
Rearranging the equation to solve for the current gives: $$ I = \frac{B \cdot l}{N \cdot \mu_0} $$
- Plug in Known Values
Substitute all known values ($B$, $l$, $N$, and $\mu_0$) into the equation. Calculate to find the value of current $I$.
- Calculate the Result
Perform the calculations step-by-step, ensuring to follow the order of operations carefully.
The value of the current needed in the coil is given by:
$$ I = \frac{B \cdot l}{N \cdot \mu_0} $$
Substituting the actual values will yield the numerical answer for $I$.
More Information
The current $I$ in the coil is crucial for achieving the desired magnetic flux density. In magnetic circuits, the concept of magnetomotive force (MMF) plays an essential role in determining how much current is needed to produce a certain magnetic field.
Tips
- Not converting units properly before calculation (e.g., ensuring lengths are in meters).
- Forgetting to include the permeability ($\mu_0$) in the calculations or using the wrong value.
- Miscalculating the number of turns of the coil.