The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The resistivity can be found in textbooks. Pure water has a resistivity value of 2.6 × 10⁵ m, while sea (salt) water has a resistivity of about 0.22 m. It is immediately clear that very little current can be transmitted through pure water. The electric field E can be approximated by the potential difference across the electrodes divided by the distance between the electrodes:

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The resistivity can be found in textbooks. Pure water has a resistivity value of 2.6 × 10⁵ m, while sea (salt) water has a resistivity of about 0.22 m. It is immediately clear that very little current can be transmitted through pure water. The electric field E can be approximated by the potential difference across the electrodes divided by the distance between the electrodes:

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The resistivity can be found in textbooks. Pure water has a resistivity value of 2.6 × 10⁵ m, while sea (salt) water has a resistivity of about 0.22 m. It is immediately clear that very little current can be transmitted through pure water. The electric field E can be approximated by the potential difference across the electrodes divided by the distance between the electrodes:

Therefore, the current in the channel is given by

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The resistivity can be found in textbooks. Pure water has a resistivity value of 2.6 × 10⁵ m, while sea (salt) water has a resistivity of about 0.22 m. It is immediately clear that very little current can be transmitted through pure water. The electric field E can be approximated by the potential difference across the electrodes divided by the distance between the electrodes:

Therefore, the current in the channel is given by

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The resistivity can be found in textbooks. Pure water has a resistivity value of 2.6 × 10⁵ m, while sea (salt) water has a resistivity of about 0.22 m. It is immediately clear that very little current can be transmitted through pure water. The electric field E can be approximated by the potential difference across the electrodes divided by the distance between the electrodes:

Therefore, the current in the channel is given by

Inserting this in Eq. (2), and assuming that the current and the magnetic field are perpendicular to each other, gives the force produced,

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The resistivity can be found in textbooks. Pure water has a resistivity value of 2.6 × 10⁵ m, while sea (salt) water has a resistivity of about 0.22 m. It is immediately clear that very little current can be transmitted through pure water. The electric field E can be approximated by the potential difference across the electrodes divided by the distance between the electrodes:

Therefore, the current in the channel is given by

Inserting this in Eq. (2), and assuming that the current and the magnetic field are perpendicular to each other, gives the force produced,

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The resistivity can be found in textbooks. Pure water has a resistivity value of 2.6 × 10⁵ m, while sea (salt) water has a resistivity of about 0.22 m. It is immediately clear that very little current can be transmitted through pure water. The electric field E can be approximated by the potential difference across the electrodes divided by the distance between the electrodes:

Therefore, the current in the channel is given by

Inserting this in Eq. (2), and assuming that the current and the magnetic field are perpendicular to each other, gives the force produced,

Note that the quantity L in Eq. (2) is the current path and is equivalent to the height (H) in this geometry because the current path is vertical. Therefore, to first order, the force produced does not depend on the channel height (H) but increases with longer channels () and wider channels (W). This conclusion is only valid where the magnetic field is constant along entire the channel. Any part of the channel that is outside of the influence of the magnets will not add to the force.

In order to optimize the force, the exact behavior of the magnetic field strength (B) as a function of magnet separation (W) must be determined. A gaussmeter was used to take measurements of the magnetic field strength between two neodymium-iron-boron (Nd-Fe-B) magnets that were positioned side by side. The results, displayed in Fig. 5, show that the magnetic field decreases as the magnet separation increases. The results were curve-fitted and integrated to determine the average magnetic field as a function of the distance between the magnets. The result is shown in Fig. 6. Note: The magnetic field is now given in teslas and the separation in meters in order to allow substitution for the magnetic field strength (B) directly into Eq. (7). The average magnetic field decreases almost inversely proportional to distance. Inserting the curvefit for the results of Fig. 6 into Eq. (7) yields an empirical force function that is proportional to a cubic function of W:

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The resistivity can be found in textbooks. Pure water has a resistivity value of 2.6 × 10⁵ m, while sea (salt) water has a resistivity of about 0.22 m. It is immediately clear that very little current can be transmitted through pure water. The electric field E can be approximated by the potential difference across the electrodes divided by the distance between the electrodes:

Therefore, the current in the channel is given by

Inserting this in Eq. (2), and assuming that the current and the magnetic field are perpendicular to each other, gives the force produced,

Note that the quantity L in Eq. (2) is the current path and is equivalent to the height (H) in this geometry because the current path is vertical. Therefore, to first order, the force produced does not depend on the channel height (H) but increases with longer channels () and wider channels (W). This conclusion is only valid where the magnetic field is constant along entire the channel. Any part of the channel that is outside of the influence of the magnets will not add to the force.

In order to optimize the force, the exact behavior of the magnetic field strength (B) as a function of magnet separation (W) must be determined. A gaussmeter was used to take measurements of the magnetic field strength between two neodymium-iron-boron (Nd-Fe-B) magnets that were positioned side by side. The results, displayed in Fig. 5, show that the magnetic field decreases as the magnet separation increases. The results were curve-fitted and integrated to determine the average magnetic field as a function of the distance between the magnets. The result is shown in Fig. 6. Note: The magnetic field is now given in teslas and the separation in meters in order to allow substitution for the magnetic field strength (B) directly into Eq. (7). The average magnetic field decreases almost inversely proportional to distance. Inserting the curvefit for the results of Fig. 6 into Eq. (7) yields an empirical force function that is proportional to a cubic function of W:

The Physics Teacher, Vol. 42, No. 7, pp. 410–415, October 2004
©2004 American Association of Physics Teachers. All rights reserved.

The dimensions of all components can be chosen as desired. Generally, the size of the boat is simply chosen so it is large enough to float the battery stack. The water channel is as long as the boat, and the channel height is chosen to match the height of available magnets. The width of the channel, however (shown in Fig. 4), allows for some optimization. Since the magnetic field diminishes as the distance from the magnets (width of the channel) increases, it appears that in order to maximize the magnetic field strength, a thin channel should be chosen. This approach is valid if the current in Eq. (2) can be independently controlled. Since our boat will be powered with batteries, however, the voltage will be constant. The current, therefore, will depend on the electrical resistance of the channel. This in turn depends on the width of the channel.

Figure 4.

In order to determine the dependence of the current on the channel dimensions, it is necessary to examine the transmission characteristics (i.e., resistances) of an electrical conductor. The current flowing from the top electrode to the bottom electrode can be expressed in terms of the current density:

where J is the current density (A/m²) and A is the cross-sectional area (W). The current density can be written in terms of the electric field E and the conductivity , or resistivity of the water:

The resistivity can be found in textbooks. Pure water has a resistivity value of 2.6 × 10⁵ m, while sea (salt) water has a resistivity of about 0.22 m. It is immediately clear that very little current can be transmitted through pure water. The electric field E can be approximated by the potential difference across the electrodes divided by the distance between the electrodes:

Therefore, the current in the channel is given by

Inserting this in Eq. (2), and assuming that the current and the magnetic field are perpendicular to each other, gives the force produced,

Note that the quantity L in Eq. (2) is the current path and is equivalent to the height (H) in this geometry because the current path is vertical. Therefore, to first order, the force produced does not depend on the channel height (H) but increases with longer channels () and wider channels (W). This conclusion is only valid where the magnetic field is constant along entire the channel. Any part of the channel that is outside of the influence of the magnets will not add to the force.

In order to optimize the force, the exact behavior of the magnetic field strength (B) as a function of magnet separation (W) must be determined. A gaussmeter was used to take measurements of the magnetic field strength between two neodymium-iron-boron (Nd-Fe-B) magnets that were positioned side by side. The results, displayed in Fig. 5, show that the magnetic field decreases as the magnet separation increases. The results were curve-fitted and integrated to determine the average magnetic field as a function of the distance between the magnets. The result is shown in Fig. 6. Note: The magnetic field is now given in teslas and the separation in meters in order to allow substitution for the magnetic field strength (B) directly into Eq. (7). The average magnetic field decreases almost inversely proportional to distance. Inserting the curvefit for the results of Fig. 6 into Eq. (7) yields an empirical force function that is proportional to a cubic function of W:

The function F versus magnet separation W is graphed in Fig. 7. The results show that an optimum does indeed exist. As the magnet separation increases from zero, the force also increases. It must increase because the conducting area is increasing [Eq. (3)]. The average magnetic field is decreasing, but not as fast as the conducting area is increasing. The result is that maximum is reached between 16 and 18 mm, after which the force decreases due to the diminishing magnetic field. The MHD boat water channel width (W) in the present study was therefore chosen to be 17 mm. These results are specific to the magnets chosen, but a similar optimum should exist for any pair of magnets. Generally, the optimum separation width between the magnets should be about the width of the pole side of the magnet, 1.2 cm in the present case.

Figure 5.

Figure 6.

Figure 7.