Problem 83 - University Physics with Modern Physics

Question

The solenoid is removed from the enclosure and then used in a location where the earth's magnetic field is 50 \(\mu\)T and points horizontally. A sample of bacteria is placed in the center of the solenoid, and the same current is applied that produced a magnetic field of 150 \(\mu\)T in the lab. Describe the field experienced by the bacteria: The field (a) is still 150 \(\mu\)T; (b) is now 200 \(\mu\)T; (c) is between 100 and 200 \(\mu\)T, depending on how the solenoid is oriented; (d) is between 50 and 150 \(\mu\)T, depending on how the solenoid is oriented.

Step-by-Step Solution

Answer

The field is between 100 and 200 \( \mu T \), depending on orientation.

Step 1: Understand the Initial Conditions

We initially have a solenoid that, when placed in a lab environment, produces a magnetic field of 150 \( \mu T \). The solenoid uses a specific current to generate this field strength.

Step 2: Recognize Environmental Change

The solenoid is moved to a new location where there is an existing horizontal earth's magnetic field of 50 \( \mu T \). The objective is to determine the total magnetic field experienced at the solenoid's center when the same current that produced 150 \( \mu T \) in the lab is applied in this new setting.

Step 3: Calculate Magnetic Field Components

The solenoid generates a magnetic field of 150 \( \mu T \) along its axis due to the applied current. The ambient earth's magnetic field contributes an additional 50 \( \mu T \), which is horizontal. These fields will combine vectorially depending on the angle of the solenoid with respect to the earth's field.

Step 4: Analyze Solenoid Orientation

If the solenoid is oriented such that its axis is parallel to the earth's field, the total field is the vector sum: 150 \( \mu T \) + 50 \( \mu T \) = 200 \( \mu T \). If perpendicular, only the solenoid's field is effective in its direction, so the magnitude between 150 \( \mu T \) and 50 \( \mu T \) applies, depending on other angular orientations.

Step 5: Conclude the Range of Fields

Thus, the total magnetic field can range from 100 \( \mu T \) (obtained when effects reduce due to orientation) to 200 \( \mu T \) (maximum when aligned). Hence, based on orientation, the field is between 100 and 200 \( \mu T \).

Key Concepts

Solenoid

A solenoid is a coil of wire that produces a magnetic field when an electric current passes through it. This field is primarily directed along the axis of the solenoid, similar to how a bar magnet works. The strength of the magnetic field generated by a solenoid depends on several factors:

• The number of turns in the wire coil - More turns result in a stronger field.
• The current flowing through the coil - Higher currents produce stronger fields.
• The presence of a core material - Inserting a ferromagnetic core can significantly enhance the field strength.

Solenoids are essential in creating controlled magnetic environments, which can be useful in experiments involving magnetic fields, like the bacteria experiment here. The field inside a solenoid is uniform and primarily concentrated along the center axis, making it ideal for precise applications.

Vector Addition

Vector addition is a method to combine multiple vectors to find a resultant vector, which represents the collective effect of the vectors being added. In our context, we have two magnetic field vectors:

• The field generated by the solenoid (150 \( \mu T \)
• The Earth's magnetic field (50 \( \mu T \))

These fields combine based on their orientations. If the fields are parallel, they add directly, leading to a stronger total field. If perpendicular, the resultant field is calculated using Pythagorean theorem: \[ B_{total} = \sqrt{(150 \mu T)^2 + (50 \mu T)^2} \]Where angles between the vectors are more nuanced, vector components are resolved along common axes and then added to find the total field experienced in different orientations.

Magnetic Field Orientation

The orientation of a magnetic field describes the direction and alignment of its lines of force. For a solenoid, this orientation is along its length, from its north to its south pole. When another field, like Earth's, interacts, its orientation can influence total field strength.
In the problem scenario, shifting the solenoid orientation changes the vector relationship between its field and Earth's field:

• Aligned (parallel): The two fields add up directly to produce a total of 200 \( \mu T \).
• Perpendicular: Each field contributes separately, not necessarily summing directly, which can reduce to midway strength.
• At arbitrary angles: The effective field value could fall anywhere depending on the cosine of the angle between the solenoid's field and Earth's field.

Understanding these principles helps in predicting how the orientation affects the field's magnitude experienced at a point.

Earth's Magnetic Field

Earth's magnetic field is a vast and complex magnetic field that extends from the planet's interior out into space. It resembles the field generated by a giant bar magnet tilted about 11 degrees from Earth's rotational axis. The strength of Earth's field varies across different locations but is typically around 50 \( \mu T \) in most horizontal components.
In the exercise, the horizontal Earth's field presents a constant magnetic field added to the solenoid's. This field is significant because it provides a reference or baseline field that interacts with the solenoid's field. It's essential to consider Earth's magnetic influence when calculating resultant fields in experiments involving external magnetic sources, as it can alter the effective field experienced by objects within these regions.