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The magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. Inside a solenoid, the magnetic field is quite uniform and can be calculated using the formula: \[ B_{\text{solenoid}} = \frac{\text{μ}_0 n I_{\text{solenoid}}}{2} \].\( \text{μ}_0 \) represents the permeability of free space, which is \(4π × 10^{-7} \text{ H/m}\).The direction of the magnetic field inside a solenoid follows the right-hand rule: if you wrap the fingers of your right hand in the direction of the current through the coils, your thumb will point in the direction of the magnetic field.Magnetic fields can be superimposed using the principle known as the superposition principle, which means the resultant field is the vector sum of all individual fields.

A solenoid is a coil of wire designed to create a strong, uniform magnetic field inside the coil. The key parameters to consider when calculating the magnetic field of a solenoid include the number of turns per unit length (\(n\)) and the current passing through the solenoid (\(I\)).The formula used to calculate the magnetic field in a solenoid is interesting because it shows that the field is proportional to the number of turns per unit length and the current.In the given exercise, the solenoid has 10.0 turns/cm. The number of turns per meter is therefore \(10.0 \text {turns/cm} × 100 = 1000 \text { turns/m} \).To calculate the field inside the solenoid, substitute values into the formula: \[ B_{\text{solenoid}} = \frac{4π × 10^{-7} \times 1000 \times 20 \times 10^{-3}}{2} \text{ T} \]. This yields a consistent internal magnetic field of approximately \(2.51327 \times 10^{-5} \text{ T}\).

Ampere's Law is one of Maxwell's equations that relates the magnetic field around a closed loop to the electric current passing through the loop. Mathematically, it is expressed as: \[ ∮ B \, • \, dl = \text{μ}_0 I_{\text{enc}} \].This means that the line integral of the magnetic field \(B\) around a closed path is equal to the permeability of free space \(\text{μ}_0\) times the total current \(I_{\text{enc}}\) enclosed by the path.Ampere's Law is particularly useful for calculating the magnetic field inside a solenoid because, by symmetry, the field strength can be considered uniform inside it.In the specific case of the solenoid, the magnetic field inside is directly proportional to the current flowing through the solenoid and the number of turns per unit length. This simplifies our calculations and allows us to use the previously mentioned formula for the magnetic field inside the solenoid.

The superposition principle states that when multiple magnetic fields are present, the total magnetic field at any point is the vector sum of the individual fields. This principle is a cornerstone of electromagnetism.In the context of the exercise, the magnetic fields due to the solenoid and the straight conductor must be superimposed to find the resultant field. The magnetic field inside the solenoid is \( B_{\text{solenoid}} = 2.51327 \times 10^{-5} \text{T} \), and the magnetic field from a straight conductor at a distance r is \[ B_{\text{wire}} = \frac{4π × 10^{-7} \times 6}{2πr} \].To find the point where these fields combine to form a resultant field at 45° to the axis of the solenoid, we equate them and solve for r:\[ 2.51327 \times 10^{-5} = \frac{12×10^{-7}}{r} \]. Solving this, we get \( r ≈ 4.77 \text{cm} \).This illustrates the powerful utility of the superposition principle in solving complex magnetic field scenarios.