Question

A \(12.0-\mathrm{V}\) battery is connected to a \(3.00-\Omega\) resistor in a rigid rectangular loop of wire measuring \(3.00 \mathrm{~m}\) by \(1.00 \mathrm{~m}\). As shown in the figure, a length \(\ell=1.00 \mathrm{~m}\) of wire at the end of the loop extends into a \(2.00 \mathrm{~m}\) by \(2.00 \mathrm{~m}\) region with a magnetic field of magnitude \(5.00 \mathrm{~T}\) directed into the page. What is the net force on the loop?

Short answer

Question: Calculate the net force on the loop when given a 12V battery, 3Ω resistor, magnetic field strength of 5T, short segment length of 1m, and long segment length of 1m inside the magnetic field. Answer: Using the given parameters, calculate the current flowing through the loop, and then find the magnetic forces experienced by the short segment and the long segment. Finally, determine the net force on the loop by subtracting the forces due to the parallel segments and considering only the forces inside the magnetic field.

Step-by-step solution

Calculate the current

Using Ohm's Law, we can calculate the current flowing through the loop. Given, Voltage (\(V\)) = 12V, and resistance (\(R\)) = 3Ω. The current (\(I\)) can be calculated as: \(I = \frac{V}{R}\)

Calculate the magnetic force on the short segment

Now, let's calculate the magnetic force on the short segment that's inside the magnetic field. The length of this segment is \(l=1\text{ m}\). The force on this segment due to the magnetic field is given by: \(F_1 = B\cdot I\cdot l\) where \(B\) is the magnetic field (\(5\text{ T}\)) and \(I\) is the current calculated in step 1.

Calculate the magnetic force on the long segment

Next, calculate the magnetic force on the long segment that's inside the magnetic field. The length of this segment is \(l'=1\text{ m}\) (the length entering the field region). The force on this segment due to the magnetic field is given by: \(F_2 = B\cdot I\cdot l'\)

Determine the net force on the loop

Since the force on the other two segments (opposite direction) and the forces on the battery and resistor are not considered in the magnetic field, the net force can be determined by the force on the parallel sides, pointing in opposite directions and perpendicular to the wire inside the magnetic field. As \(F_1\) and \(F_2\) are acting in opposite directions, the net force is given by: \(F_\text{net} = F_1 - F_2\) Plug the values found from step 1, 2 and 3 into the equation above to find the net force.

Key concepts

Ohm's Law

Understanding Ohm's Law is essential when working with electric circuits. At the heart of this principle is a simple formula: \[\begin{equation}I = \frac{V}{R}\end{equation}\]where

• \[\begin{equation}I\end{equation}\] represents the current in amperes (A),
• \[\begin{equation}V\end{equation}\] is the voltage in volts (V), and
• \[\begin{equation}R\end{equation}\] denotes the resistance in ohms (\(\Omega\)).

In our exercise, Ohm's Law helps us determine the electric current flowing through a resistor. We take the voltage provided by a battery and divide it by the resistance of the loop. For instance, with a 12-volt battery and a resistance of 3 ohms, the circuit's current is 4 amperes. It's important to recognize that according to Ohm's Law, if we increase the resistance while keeping the voltage constant, the current would decrease, and vice versa.

Magnetic Field

A magnetic field can be envisioned as a space where magnetic forces are exerted, represented by field lines. Common units include the tesla (T) or the gauss (G), where 1 T equals 10,000 G:\[\begin{equation}1 \text{T} = 10,000 \text{G}\end{equation}\]A magnetic field exerts a force on charged particles moving through it, such as the electrons flowing in the current through our wire loop. This force always acts perpendicular to the velocity of the charged particles and the direction of the magnetic field. In our problem, the magnetic field strength is 5.00 T and it's directed into the page, affecting only the part of the wire that lies within the region of this field.

Lorentz Force

The Lorentz force is the combined electric and magnetic force on a point charge due to electromagnetic fields. When dealing purely with the impact of a magnetic field on a current-carrying wire, we can focus on the magnetic component of the Lorentz force, which can be calculated using the formula:\[\begin{equation}F = B \times I \times l\end{equation}\]where

• \[\begin{equation}F\end{equation}\] is the magnetic force in newtons (N),
• \[\begin{equation}B\end{equation}\] is the magnetic field strength in teslas (T),
• \[\begin{equation}I\end{equation}\] is the current in amperes (A), and
• \[\begin{equation}l\end{equation}\] is the length of wire in the field in meters (m).

In the context of our textbook exercise, the Lorentz force tells us how the magnetic field affects the specific segments of the wire carrying the electric current. Calculating these forces is key to understanding the net force acting on the entire loop.

Net Force Calculation

To calculate the net force in a system, we tally up all the individual forces acting upon the object. Forces can be vector quantities, which means they have both magnitude and direction. When accounting for the direction, we can encounter scenarios where forces cancel each other out or add up to create a resultant force.In our exercise, we apply this by considering the forces acting on different segments of the loop due to the magnetic field. Because the segments of the wire within the magnetic field are parallel and the magnetic forces act in opposite directions, one might expect a straightforward subtraction:\[\begin{equation}F_{\text{net}} = F_{1} - F_{2}\end{equation}\]However, careful analysis reveals that both segments experience forces of equal magnitudes but opposite directions, resulting in no net force, hence \[\begin{equation}F_{\text{net}} = 0\end{equation}\]Understanding net force is crucial for problems involving multiple forces and helps us predict the motion of the loop in response to these forces.